abs-anab-topics-2.pdf

TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II: DECOMPOSITION GROUPS AND ENDOMORPHISMS Shinichi Mochizuki October 2013 Abstract. The present paper, which forms the second part of a three-part series in which we study absolute anabelian geometry from an algorithmic point of view, focuses on the study of the closely related notions of decomposition groups and endomorphisms in this anabelian context. We begin by studying an abstract combinatorial analogue of the algebro-geometric notion of a stable polycurve [i.e., a “successive extension of families of stable curves”] and showing that the “geometry of log divisors on stable polycurves” may be extended, in a purely group-theoretic fashion, to this abstract combinatorial analogue; this leads to various anabelian results concerning conﬁguration spaces. We then turn to the study of the absolute pro-Σ anabelian geometry of hyperbolic curves over mixed-characteristic local ﬁelds, for Σ a set of primes of cardinality ≥ 2 that contains the residue characteristic of the base ﬁeld. In particular, we prove a certain “pro-p resolution of nonsingularities” type result, which implies a “conditional” anabelian result to the eﬀect that the condition, on an isomorphism of arithmetic fundamental groups, of preservation of decomposition groups of “most” closed points implies that the isomorphism arises from an isomorphism of schemes — i.e., in a word, “point-theoreticity implies geometricity”; a “non-conditional” version of this result is then obtained for “pro- curves” obtained by removing from a proper curve some set of closed points which is “p-adically dense in a Galois-compatible fashion”. Finally, we study, from an algorithmic point of view, the theory of Belyi and elliptic cuspidalizations, i.e., group-theoretic reconstruction algorithms for the arithmetic fundamental group of an open subscheme of a hyperbolic curve that arise from consideration of certain endomorphisms determined by Belyi maps and endomorphisms of elliptic curves. Contents: §0. Notations and Conventions §1. A Combinatorial Analogue of Stable Polycurves §2. Geometric Uniformly Toral Neighborhoods §3. Elliptic and Belyi Cuspidalizations Introduction In the present paper, which forms the second part of a three-part series, we continue our discussion of various topics in absolute anabelian geometry from 2000 Mathematical Subject Classiﬁcation. Primary 14H30; Secondary 14H25. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI a “group-theoretic algorithmic” point of view, as discussed in the Introduction to [Mzk15]. The topics presented in the present paper center around the following two themes: (A) [the subgroups of arithmetic fundamental groups constituted by] de- composition groups of subvarieties of a given variety [such as closed points, divisors] as a crucial tool that leads to absolute anabelian results; (B) “hidden endomorphisms” — which may be thought of as “hidden symmetries” — of hyperbolic curves that give rise to various absolute anabelian results. In fact, “decomposition groups” and “endomorphisms” are, in a certain sense, re- lated notions — that is to say, the monoid of “endomorphisms” of a variety may be thought of as a sort of “decomposition group of the generic point”! With regard to the theme (B), we recall that the endomorphisms of an abelian variety play a fundamental role in the theory of abelian varieties [e.g., elliptic curves!]. Unlike abelian varieties, hyperbolic curves [say, in characteristic zero] do not have suﬃcient “endomorphisms” in the literal, scheme-theoretic sense to form the basis for an interesting theory. This diﬀerence between abelian varieties and hyperbolic curves may be thought of, at a certain level, as reﬂecting the dif- ference between linear Euclidean geometries and non-linear hyperbolic geometries. From this point of view, it is natural to search for “hidden endomorphisms” that are, in some way, related to the intrinsic non-linear hyperbolic geometry of a hy- perbolic curve. Examples [that appear in previous papers of the author] of such “hidden endomorphisms” — which exhibit a remarkable tendency to be related [for instance, via some induced action on the arithmetic fundamental group] to some sort of “anabelian result” — are the following: (i) the interpretation of the automorphism group P SL 2 (R) of the uni- versal covering of a hyperbolic Riemann surface as an object that gives rise to a certain “Grothendieck Conjecture-type result” in the “geometry of categories” [cf. [Mzk11], Theorem 1.12]; (ii) the interpretation of the theory of Teichmüller mappings [a sort of endomorphism — cf. (iii) below] between hyperbolic Riemann surfaces as a “Grothendieck Conjecture-type result” in the “geometry of categories” [cf. [Mzk11], Theorem 2.3]; (iii) the use of the endomorphisms constituted by Frobenius liftings — in the form of p-adic Teichmüller theory — to obtain the absolute anabelian result constituted by [Mzk6], Corollary 3.8; (iv) the use of the endomorphism rings of Lubin-Tate groups to obtain the absolute anabelian result constituted by [Mzk15], Corollaries 3.8, 3.9. The main results of the present paper — in which both themes (A) and (B) play a central role — are the following: (1) In §1, we develop a purely combinatorial approach to the algebro-geometric notion of a stable polycurve [cf. [Mzk2], Deﬁnition 4.5]. This approach may TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II be thought of as being motivated by the purely combinatorial approach to the notion of a stable curve given in [Mzk13]. Moreover, in §1, we apply the theory of [Mzk13] to give, in eﬀect, “group-theoretic algorithms” for reconstructing the “abstract combinatorial analogue” of the geome- try of the various divisors — in the form of inertia and decomposition groups — associated to the canonical log structure of a stable polycurve [cf. Theorem 1.7]. These techniques, together with the theory of [MT], give rise to various relative and absolute anabelian results concerning conﬁguration spaces associated to hyperbolic curves [cf. Corollaries 1.10, 1.11]. Relative to the discussion above of “hidden en- domorphisms”, we observe that such conﬁguration spaces may be thought of as representing a sort of “tautological endomorphism/correspondence” of the hyperbolic curve in question. (2) In §2, we study the absolute pro-Σ anabelian geometry of hyperbolic curves over mixed-characteristic local ﬁelds, for Σ a set of primes of car- dinality ≥ 2 that contains the residue characteristic of the base ﬁeld. In particular, we show that the condition, on an isomorphism of arithmetic fundamental groups, of preservation of decomposition groups of “most” closed points implies that the isomorphism arises from an isomorphism of schemes — i.e., in a word, “point-theoreticity implies geometricity” [cf. Corollary 2.9]. This condition may be removed if one works with “pro- curves” obtained by removing from a proper curve some set of closed points which is “p-adically dense in a Galois-compatible fashion” [cf. Corollary 2.10]. The key technical result that underlies these anabelian results is a certain “pro-p resolution of nonsingularities” type result [cf. Lemma 2.6; Remark 2.6.1; Corollary 2.11] — i.e., a result reminiscent of the main [proﬁnite] results of [Tama2]. This technical result allows one to apply the theory of uniformly toral neighborhoods developed in [Mzk15], §3. Relative to the discussion above of “hidden endomorphisms”, this technical result is interesting [cf., e.g., (iii) above] in that one central step of the proof of the technical result is quite similar to the well-known classical argument that implies the nonexistence of a Frobenius lifting for stable curves over the ring of Witt vectors of a ﬁnite ﬁeld [cf. Remark 2.6.2]. (3) In §3, we re-examine the theory of [Mzk8], §2, for reconstructing the decomposition groups of closed points from the point of view of the present series of developing “group-theoretic algorithms”. In particular, we observe that these group-theoretic algorithms allow one to use Belyi maps and endomorphisms of elliptic curves to con- struct [not only decomposition groups of closed points, but also] “cuspidalizations” 3 4 SHINICHI MOCHIZUKI [i.e., the full arithmetic fundamental groups of the open subschemes ob- tained by removing various closed points — cf. the theory of [Mzk14]] as- sociated to various types of closed points [cf. Corollaries 3.3, 3.4, 3.7, 3.8]. Relative to the discussion above of “hidden endomorphisms”, the theory of Belyi and elliptic cuspidalizations given in §3 illustrates quite explicitly how endomorphisms [arising from Belyi maps or endomorphisms of elliptic curves] can give rise to group-theoretic reconstruction algorithms. Finally, we remark that although the “algorithmic approach” to stating anabelian results is not carried out very explicitly in §1, §2 [by comparison to §3 or [Mzk15]], the translation into “algorithmic language” of the more traditional “Grothendieck Conjecture-type” statements of the main results of §1, §2 is quite routine. [Here, it should be noted that the results of §1 that depend on “Uchida’s theorem” — i.e., Theorem 1.8, (ii); Corollary 1.11, (iv) — constitute a notatable exception to this “remark”, an exception that will be discussed in more detail in [Mzk16] — cf., e.g., [Mzk16], Remark 1.9.5.] That is to say, this translation was not carried out explicitly by the author solely because of the complexity of the algorithms implicit in §1, §2, i.e., not as a result of any substantive mathematical obstacles. Acknowledgements: I would like to thank Akio Tamagawa and Yuichiro Hoshi for many helpful discussions concerning the material presented in this paper and Emmanuel Lepage for a suggestion that led to Remark 2.11.1, (i). TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 5 Section 0: Notations and Conventions We shall continue to use the “Notations and Conventions” of [Mzk15], §0. In addition, we shall use the following notation and conventions: Topological Groups: Let G be a topologically ﬁnitely generated, slim proﬁnite group. Thus, G admits a basis of characteristic open subgroups. Any such basis determines a proﬁnite topology on the groups Aut(G), Out(G). If ρ : H → Out(G) is any continuous homomorphism of proﬁnite groups, then we denote by out G H the proﬁnite group obtained by pulling back the natural exact sequence of proﬁnite groups 1 → G → Aut(G) → Out(G) → 1 via ρ. Thus, we have a natural exact out sequence of proﬁnite groups 1 → G → G H → H → 1. Semi-graphs: Let Γ be a connected semi-graph [cf., e.g., [Mzk9], §1, for a review of the theory of semi-graphs]. We shall refer to the [possibly inﬁnite] dimension over Q of the singular homology module H 1 (Γ, Q) as the loop-rank lp-rk(Γ) of Γ. We shall say that Γ is loop-ample if, for any edge e of Γ, the semi-graph obtained from Γ by removing e remains connected. We shall say that Γ is untangled if every closed edge of Γ abuts to two distinct vertices [cf. [Mzk9], §1]. We shall say that Γ is edge-paired (respectively, edge-even) if Γ is untangled, and, moreover, for any two [not necessarily distinct!] vertices v, v of Γ, the set of edges e of Γ such that e abuts to a vertex w of Γ if and only if w ∈ {v, v } is either empty or of cardinality ≥ 2 (respectively, empty or of even cardinality). [Thus, one veriﬁes immediately that if Γ is edge-paired (respectively, edge-even), then it is loop-ample (respectively, edge-paired).] We shall refer to as a simple path in Γ any connected subgraph γ ⊆ Γ such that the following conditions are satisﬁed: (a) γ is a ﬁnite tree that has at least one edge; (b) given any vertex v of γ, there exist at most two branches of edges of γ that abut to v. Thus, [one veriﬁes easily that] a simple path γ has precisely two vertices v such that there exists precisely one branch of an edge of γ that abuts to v; we shall refer to these two vertices as the terminal vertices of the simple path γ. If γ, γ are simple paths in Γ such that the terminal vertices of γ, γ coincide, then we shall say that γ, γ are co-terminal. Log Schemes: We shall often regard a scheme as a log scheme equipped the trivial log struc- ture. Any ﬁber product of fs [i.e., ﬁne, saturated] log schemes is to be taken in the category of fs log schemes. In particular, the underlying scheme of such a product is ﬁnite over, but not necessarily isomorphic to, the ﬁber product of the underlying schemes. 6 SHINICHI MOCHIZUKI Curves: We shall refer to a hyperbolic orbicurve X as semi-elliptic [i.e., “of type (1, 1) ± ” in the terminology of [Mzk12], §0] if there exists a ﬁnite étale double covering Y → X, where Y is a once-punctured elliptic curve, and the covering is given by the stack-theoretic quotient of Y by the “action of ±1” [i.e., relative to the group operation on the elliptic curve given by the canonical compactiﬁcation of Y ]. For i = 1, 2, let X i be a hyperbolic orbicurve over a ﬁeld k i . Then we shall say that X 1 , X 2 are isogenous [cf. [Mzk14], §0] if there exists a hyperbolic orbicurve X over a ﬁeld k, together with ﬁnite étale morphisms X → X i , for i = 1, 2. Note that in this situation, the morphisms X → X i induce ﬁnite separable inclusions of ﬁelds k i → k. [Indeed, this follows immediately from the easily veriﬁed fact that × every subgroup G ⊆ Γ(X, O X ) such that G {0} determines a ﬁeld is necessarily contained in k × .] We shall use the term stable log curve as it was deﬁned in [Mzk9], §0. Let X log → S log be a stable log curve over an fs log scheme S log , where S = Spec(k) for some ﬁeld k; k a separable closure of k. Then we shall refer to as the loop-rank lp-rk(X log ) [or lp-rk(X)] of X log [or X] the loop-rank of the dual graph of X log × k k [or X × k k]. We shall say that X log [or X] is loop-ample (respectively, untangled; edge-paired; edge-even) if the dual semi-graph with compact structure [cf. [Mzk5], Appendix] of X log × k k is loop-ample (respectively, untangled; edge-paired; edge-even) [as a connected semi-graph]. We shall say that X log [or X] is sturdy if the normalization of every irreducible component of X is of genus ≥ 2 [cf. [Mzk13], Remark 1.1.5]. Observe that for any prime number l invertible on S, there exist an fs log scheme T log over S log , where T = Spec(k ), for some ﬁnite separable extension k of k, and a connected Galois log admissible covering Y log → X log × S log T log [cf. [Mzk1], §3] of degree a power of l such that Y log is sturdy and edge-paired [hence, in particular, untangled and loop-ample]; if, moreover, l = 2, then one may also take Y log to be edge-even. [Indeed, to verify this observation, we may assume that k = k. Then note that any hyperbolic curve U over k admits a connected ﬁnite étale Galois covering V → U of degree a power of l such that V is of genus ≥ 2 and ramiﬁed with ramiﬁcation index l 2 at each of the cusps of V — cf. the discussion at the end of the present §0. Thus, by gluing together such coverings at the nodes of X, one concludes that there exists a connected Galois log admissible covering Z 1 log → X log × S log T log of degree a power of l which is totally ramiﬁed over every node of X with ramiﬁcation index l 2 such that every irreducible component of Z 1 is of genus ≥ 2 — i.e., Z 1 log is sturdy. Next, observe that there exists a connected Galois log admissible covering Z 2 log → Z 1 log of degree a power of l that arises from a covering of the dual graph of Z 1 such that Z 2 log is untangled [and still sturdy]. Finally, observe that there exists a connected Galois log admissible covering Z 3 log → Z 2 log of degree a positive power of l which restricts to a connected ﬁnite étale covering over every irreducible component of Z 2 [hence is unramiﬁed at the nodes] such that Z 3 log is edge-paired [and still sturdy and untangled] for arbitrary def l and edge-even when l = 2. Thus, we may take Y log = Z 3 log .] TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 7 Observe that if X is loop-ample, then for every point x ∈ X(k) which is not a unique cusp of X [i.e., either x is not a cusp or if x is a cusp, then it is not the unique cusp of X], the evaluation map H 0 (X, ω X log /S log ) → ω X log /S log | x is surjective. Indeed, by considering the long exact sequence associated to the short exact sequence 0 → ω X log /S log ⊗ O X I x → ω X log /S log → ω X log /S log | x → 0, where I x ⊆ O X is the sheaf of ideals corresponding to x, one veriﬁes immediately that it suﬃces to show that the surjection def def H x = H 1 (X, ω X log /S log ⊗ O X I x ) H = H 1 (X, ω X log /S log ) is injective. If x is a node, then the fact that X is loop-ample implies [by computing via Serre duality] that either dim k (H x ) = dim k (H) = 1 [if X has no cusps] or dim k (H x ) = dim k (H) = 0 [if X has cusps]. Thus, we may assume that x is not a node, so the surjection H x H is dual to the injection def def M = H 0 (X, O X (−D)) → M x = H 0 (X, O X (x − D)) def (⊆ N x = H 0 (X, O X (x))) — where we write D ⊆ X for the divisor of cusps of X. If x is a cusp, then it follows that D is of degree ≥ 2, and one computes easily that dim k (M) = dim k (M x ) = 0. Thus, we may assume that x is not a cusp. Write C for the irreducible component of X containing x. Now suppose that the injection M → M x is not surjective. Then it follows that dim k (N x ) = 2, and that N x determines a basepoint-free linear system on X. In particular, N x determines a morphism φ : X → P 1 k that is of ∼ degree 1 on C — i.e., determines an isomorphism C → P 1 k — and constant on the other irreducible components of X. Since X is loop-ample, it follows that the dual graph Γ of X either has no edges or admits a loop containing the vertex determined by C. On the other hand, the existence of such a loop contradicts the fact that φ ∼ determines an isomorphism C → P 1 k . Thus, we conclude that X = C ∼ = P 1 k , so D is of degree ≥ 3. But this implies that M x = 0, a contradiction. Finally, let U be a hyperbolic curve over an algebraically closed ﬁeld k and l a prime number invertible in k. Suppose that the cardinality r of the set of cusps of U is ≥ 2, and, moreover, that, if l = 2, then r is even. Then observe that it follows immediately from the well-known structure of the maximal pro-l quotient of the abelianization of the étale fundamental group of U that for every power l n of l, where n is a positive integer, there exists a cyclic covering V → U of degree l n that is totally ramiﬁed over the cusps of U . Indeed, this observation is an immediate consequence of the elementary fact that, in light of our assumptions on r, there always exist r − 1 integers prime to l whose sum is also prime to l. We shall often make use of the assumption that a stable log curve is edge-paired — or, when l = 2, edge-even — by applying the above observation to the various connected components of the complement of the cusps and nodes of the stable log curve. 8 SHINICHI MOCHIZUKI Section 1: A Combinatorial Analogue of Stable Polycurves In the present §1, we apply the theory of [Mzk13] to study a sort of purely group-theoretic, combinatorial analogue [cf. Deﬁnition 1.5 below] of the notion of a stable polycurve introduced in [Mzk2], Deﬁnition 4.5. This allows one to reconstruct the “abstract combinatorial analogue” of the “geometry of log divisors” [i.e., divisors associated to the log structure of a stable polycurve] of such a combinatorial object via group theory [cf. Theorem 1.7]. Finally, we apply the theory of [MT] to obtain various consequences of the theory of the present §1 [cf. Corollaries 1.10, 1.11] concerning the absolute anabelian geometry of conﬁguration spaces. We begin by recalling the discussion of [Mzk13], Example 2.5. Example 1.1. Stable Log Curves over a Logarithmic Point (Revisited). (i) Let k be a separably closed ﬁeld; Σ a nonempty set of prime numbers in- vertible in k; M ⊆ Q the monoid of positive rational numbers with denomina- tors invertible in k; S log (respectively, T log ) the log scheme obtained by equipping def def S = Spec(k) (respectively, T = Spec(k)) with the log structure determined by the chart N 1 → 0 ∈ k (respectively, M 1 → 0 ∈ k); T log → S log the morphism determined by the natural inclusion N → M; X log → S log def a stable log curve over S log . Thus, the proﬁnite group I S log = Aut(T log /S log ) ∼ admits a natural isomorphism I S log → Hom(Q/Z, k × ) and ﬁts into an natural exact sequence def def 1 → Δ X log = π 1 (X log × S log T log ) → Π X log = π 1 (X log ) → I S log → 1 — where we write “π 1 (−)” for the “log fundamental group” of the log scheme in parentheses [which amounts, in this case, to the fundamental group arising from the admissible coverings of X log ], relative to an appropriate choice of basepoint [cf. [Ill] for a survey of the theory of log fundamental groups]. In particular, if we write I S Σ log for the maximal pro-Σ quotient of I S log , then as abstract proﬁnite groups, Σ , where we write Z Σ for the maximal pro-Σ quotient of Z. I S Σ log ∼ = Z (ii) On the other hand, X log determines a semi-graph of anabelioids [cf. [Mzk9], Deﬁnition 2.1] of pro-Σ PSC-type [cf. [Mzk13], Deﬁnition 1.1, (i)], whose underlying semi-graph we denote by G. Thus, for each vertex v [corresponding to an irreducible component of X log ] (respectively, edge e [corresponding to a node or cusp of X log ]) of G, we have a connected anabelioid [i.e., a Galois category] G v (respectively, G e ), and for each branch b of an edge e abutting to a vertex v, we are given a morphism of anabelioids G e → G v . Then the maximal pro-Σ completion of Δ X log may be identiﬁed with the “PSC-fundamental group” Π G associated to G. Also, we recall that Π G is slim [cf., e.g., [Mzk13], Remark 1.1.3], and that the groups Aut(G), Out(Π G ) may be equipped with proﬁnite topologies in such a way that the natural morphism Aut(G) → Out(Π G ) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 9 is a continuous injection [cf. the discussion at the beginning of [Mzk13], §2], which we shall use to identify Aut(G) with its image in Out(Π G ). In particular, we obtain a natural continuous homomorphism I S log → Aut(G). Moreover, it follows imme- diately from the well-known structure of admissible coverings at nodes [cf., e.g., [Mzk1], §3.23] that this homomorphism factors through I S Σ log , hence determines a natural continuous homomorphism ρ I : I S Σ log → Aut(G). Also, we recall that each vertex v (respectively, edge e) of G determines a(n) verticial subgroup Π v ⊆ Π G (respectively, edge-like subgroup Π e ⊆ Π G ), which is well-deﬁned up to conjugation — cf. [Mzk13], Deﬁnition 1.1, (ii). Here, the edge-like subgroups Π e may be either nodal or cuspidal, depending on whether e corresponds to a node or to a cusp. If an edge e corresponds to a node (respectively, cusp), then we shall simply say that e “is” a node (respectively, cusp). (iii) Let e be a node of X. Write M e for the stalk of the characteristic sheaf of the log scheme X log at e; M S for the stalk of the characteristic sheaf of the log scheme S log at the tautological S-valued point of S. Thus, M S ∼ = N; we have a natural inclusion M S → M e , with respect to which we shall often [by abuse of notation] identify M S with its image in M e . Write σ ∈ M e for the unique generator of [the image of] M S . Then there exist elements ξ, η ∈ M e satisfying the relation ξ + η = i e · σ for some positive integer i e , which we shall refer to as the index of the node e, such that M e is generated by ξ, η, σ. Also, we shall write i Σ e for the largest positive integer j such that i e /j is a product of primes ∈ Σ and refer to i Σ e as the Σ-index of the node e. One veriﬁes easily that the set of elements {ξ, η} of M e may be characterized intrisically as the set of elements θ ∈ M e \M S such that any relation θ = n · θ + θ for n a positive integer, θ ∈ M e \M S , θ ∈ M S implies that n = 1, θ = 0. In particular, i e , i Σ e are well-deﬁned and depend only on the isomorphism class of the pair consisting of the monoid M e and the submonoid ⊆ M e given by the image of M S . Remark 1.1.1. Of course, in Example 1.1, it is not necessary to assume that k is separably closed [cf. [Mzk13], Example 2.5]. If k is not separably closed, then one must also contend with the action of the absolute Galois group of k. More generally, for the theory of the present §1, it is not even necessary to assume that an “additional proﬁnite group” acting on G arises “from scheme theory”. It is this point of view that formed the motivation for Deﬁnition 1.2 below. Deﬁnition 1.2. In the notation of Example 1.1: (i) Let ρ H : H → Aut(G) (⊆ Out(Π G )) be a continuous homomorphism of proﬁnite groups; suppose that X log is nonsingular [i.e., has no nodes]. Then we shall refer to as a [pro-Σ] PSC-extension [i.e., “pointed stable curve extension”] any extension of proﬁnite groups that is isomorphic — via an isomorphism which we shall refer to as the “structure of [pro-Σ] PSC-extension” — to an extension of the form out def 1 → Π G → Π H = (Π G H) → H → 1 10 SHINICHI MOCHIZUKI out [cf. §0 for more on the notation “ ”], which we shall refer to as the PSC- extension associated to the construction data (X log → S log , Σ, G, ρ H ). In this sit- uation, each [necessarily cuspidal] edge e of G determines [up to conjugation in def Π G ] a subgroup Π e ⊆ Π G , whose normalizer D e = N Π H (Π e ) in Π H we shall refer to as the decomposition group associated to the cusp e; we shall refer to def I e = Π e = D e Π G ⊆ D e [cf. [Mzk13], Proposition 1.2, (ii)] as the inertia group associated to the cusp e. Finally, we shall apply the terminology applied to objects associated to 1 → Π G → Π H → H → 1 to the objects associated to an arbitrary PSC-extension via its “structure of PSC-extension” isomorphism. (ii) Let ρ H : H → Aut(G) (⊆ Out(Π G )) be a continuous homomorphism of proﬁnite groups; ι : I S Σ log → H a continuous injection of proﬁnite groups with normal image such that ρ H ◦ ι = ρ I . Suppose that X log is arbitrary [i.e., X may be singular or nonsingular]. Then we shall refer to as a [pro-Σ] DPSC-extension [i.e., “degenerating pointed stable curve extension”] any extension of proﬁnite groups that is isomorphic — via an isomorphism which we shall refer to as the “structure of [pro-Σ] DPSC-extension” — to an extension of the form def out 1 → Π G → Π H = (Π G H) → H → 1 — which we shall refer to as the DPSC-extension associated to the construction data (X log → S log , Σ, G, ρ H , ι). In this situation, we shall refer to the image I ⊆ H of ι as the inertia subgroup of H and to the extension def out 1 → Π G → Π I = (Π G I) → I → 1 [so Π I = Π H × H I ⊆ Π H ] as the [pro-Σ] IPSC-extension [i.e., “inertial pointed stable curve extension”] associated to the construction data (X log → S log , Σ, G, ρ H , ι); each vertex v (respectively, edge e) of G determines [up to conjugation in Π G ] a subgroup Π v ⊆ Π G (respectively, Π e ⊆ Π G ), whose normalizer def def D v = N Π H (Π v ) (respectively, D e = N Π H (Π e )) in Π H we shall refer to as the decomposition group associated to v (respectively, e); for v arbitrary (respectively, e a node), we shall refer to the centralizer def def I v = Z Π I (Π v ) ⊆ D v (respectively, I e = Z Π I (Π e ) ⊆ D e ) as the inertia group associated to v (respectively, e). If e is a cusp of G, then we shall def refer to I e = Π e = D e Π G ⊆ D e [cf. [Mzk13], Proposition 1.2, (ii)] as the inertia group associated to the cusp e. Finally, we shall apply the terminology applied to objects associated to 1 → Π G → Π H → H → 1 to the objects associated to an arbitrary DPSC-extension via its “structure of DPSC-extension” isomorphism. Remark 1.2.1. Note that in the situation of Deﬁnition 1.2, (i) (respectively, (ii)), any open subgroup of Π H [equipped with the induced extension structure] admits a structure of [pro-Σ] PSC-extension (respectively, DPSC-extension) for appropriate TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 11 construction data that may be derived from the original construction data. On the other hand, in the situation of Deﬁnition 1.2, (ii), given an open subgroup of Π I [equipped with the induced extension structure], in order to endow this open subgroup with a structure of IPSC-extension, it may be necessary — even if, for instance, this open subgroup of Π I surjects onto I — to replace the inertia subgroup I of H by some open subgroup of I [i.e., in eﬀect, to replace the given open subgroup of Π I with the intersection of this given open subgroup with the inverse image in Π I of some open subgroup of I]. Such replacements may be regarded as a sort of abstract group-theoretic analogue of the operation of passing to a ﬁnite extension of a discretely valued ﬁeld in order to achieve a situation in which a given hyperbolic curve over the original ﬁeld has stable reduction. Remark 1.2.2. Note that in the situation of Deﬁnition 1.2, (ii), the inertia subgroup I ⊆ H is not intrinsically determined in the sense that any open subgroup of I may also serve as the inertia subgroup of H — cf. the replacement operation discussed in Remark 1.2.1. Remark 1.2.3. Recall that for l ∈ Σ, one may construct directly from G a pro-l cyclotomic character χ l : Aut(G) → Z × l [cf. [Mzk13], Lemma 2.1]. In particular, any ρ H as in Deﬁnition 1.2, (i), (ii), determines a pro-l cyclotomic character χ l | H : H → Z × l . The action ρ H is called l-cyclotomically full [cf. [Mzk13], Deﬁnition 2.3, (ii)] if the image of χ l | H is open. We shall also apply this terminology “l- cyclotomically full” to the corresponding PSC-, DPSC-extensions. In fact, it follows immediately from the ﬁrst portion of [Mzk13], Proposition 2.4, (iv), that the issue of whether or not ρ H is l-cyclotomically full depends only on the outer representation H → Out(Π G ) of H on Π G determined by ρ H . Proposition 1.3. (Basic Properties of Inertia and Decomposition Groups) In the notation of Deﬁnition 1.2, (ii): Σ . (i) If e is a cusp of G, then as abstract proﬁnite groups, I e ∼ = Z (ii) If e is a node of G, then we have a natural exact sequence 1 → Π e → Σ × Z Σ . If e abuts to vertices v, v , I e → I → 1; as abstract proﬁnite groups, I e ∼ = Z then [for appropriate choices of conjugates of the various inertia groups involved] we have inclusions I v , I v ⊆ I e , and the natural morphism I v × I v → I e is an open injective homomorphism, with image of index equal to i Σ e . ∼ (iii) If v is a vertex of G, then we have a natural isomorphism I v → I; Σ . If e is a cusp that D v Π I = I v × Π v ; as abstract proﬁnite groups, I v ∼ = Z abuts to v, then [for appropriate choices of conjugates of the various inertia and decomposition groups involved] we have inclusions I e , I v ⊆ D e Π I , and the nat- ural morphism I e × I v → D e Π I is an isomorphism; in particular, as abstract ∼ Σ Σ proﬁnite groups, D e Π I = Z × Z , and we have a natural exact sequence 1 → I e → D e Π I → I → 1. (iv) Let v, v be vertices of G. If D v D v Π I = {1}, then one of the following three [mutually exclusive] properties holds: (1) v = v ; (2) v and v are 12 SHINICHI MOCHIZUKI distinct, but adjacent [i.e., there exists a node e that abuts to v, v ]; (3) v and v are distinct and non-adjacent, but there exists a vertex v = v, v of G such that v is adjacent to v and v . Moreover, in the situation of (2), we have I v I v = {1}, [for appropriate choices of conjugates of the various inertia and decomposition D Π Π v = = I ; in the situation of (3), we have Π groups involved] D v v I e v of conjugates of I v I v = I v I v = I v I v = {1}, [for appropriate choices the various inertia and decomposition groups involved] D v D v Π I = I v . In particular, I v I v = {1} implies that v = v . (v) Let v be a vertex of G. Then D v = C Π H (I v ) = N Π H (I v ) is commen- surably terminal in Π H ; D v Π I = C Π I (I v ) = N Π I (I v ) = Z Π I (I v ) is com- mensurably terminal in Π I ; D v Π G = Π v is commensurably terminal in Π G . (vi) Let v be a vertex of G. Then the image of D v in H is open; on the other hand, if G has more than one vertex [i.e., the curve X is singular], then D v is not open in Π H . = N Π H (Π e ) is commensu- (vii) Let e be an edge of G. Then D e = C Π H (Π e ) rably terminal in Π H . If e is a node, then I e = D e Π I . (viii) Let e, e be edges of G. If D e D e Π I = {1}, then one of the following two [mutually exclusive] properties holds: (1) e = e ; (2) e and e are distinct, but abut to the same vertex v, and D e D e Π G = {1}. Moreover, in the situation of (2), [for appropriate choices of conjugates of the various inertia and decomposition groups involved] we have I v = D e D e Π I . (ix) Let e be an edge of G. Then the image of D e in H is open, but D e is not open in Π H . (x) Let τ I : I → Π I be the [outer] homomorphism that arises [by functoriality!] from a “log point” τ S ∈ X log (S log ). Let us call τ I non-verticial (respectively, non-edge-like) if τ I (I) is not contained in I v (respectively, I e ) for any vertex v (respectively, edge e) of G. Then if τ I is non-verticial and non-edge-like, then the image of τ S is the unique cusp e τ of X such that [for an appropriate choice of conjugate of D e τ ] τ I (I) ⊆ D e τ . Now suppose that the image of τ S is not a cusp. Then τ I satisﬁes the condition τ I (I) = I v τ for some vertex v τ of G [and an appropriate choice of conjugate of I v τ ] if and only if the image of τ S is a non-nodal point of the irreducible component of X corresponding to v τ ; τ I is non-verticial and satisﬁes the condition τ I (I) ⊆ I e τ for some node e τ of G [and an appropriate choice of conjugate of I e τ ] if and only if the image of τ S is the node of X corresponding to e τ . Proof. Assertion (i) follows immediately from the deﬁnitions. Next, we consider assertion (ii). Write ν ( ∼ = S) for the closed subscheme of X determined by the node of X corresponding to e; ν log for the result of equipping ν with the log structure def pulled back from X log . Thus, we obtain a natural [outer] homomorphism Π ν = π 1 (ν log ) → π 1 (X log ) = Π X log Π I . Now [in the notation of Example 1.1, (iii)] one computes easily [by considering the Galois groups of the various Kummer log étale ∼ coverings of ν log ] that we have natural isomorphisms Π ν → Hom(M e gp ⊗ Q/Z, k × ), TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 13 ∼ I S log → Hom(M S gp ⊗ Q/Z, k × ) [where “gp” denotes the groupiﬁcation of a monoid]. Moreover, if we write Π ν Π Σ ν for the maximal pro-Σ quotient of Π ν , then one veriﬁes immediately that the isomorphisms induced on maximal pro-Σ quotients by these natural isomorphisms are compatible, relative to the surjection ∼ gp gp × × Σ Σ ∼ Σ (Π Σ ν → ) Hom(M e ⊗ Q/Z, k ) ⊗ Z Hom(M S ⊗ Q/Z, k ) ⊗ Z ( → I S log ) Σ induced by the inclusion M S → M e , with the morphism Π Σ ν → I S log induced by ∼ gp Σ . the composite morphism Π ν → Π I I → I S Σ log ∼ = Hom(M S ⊗ Q/Z, k × ) ⊗ Z Σ The kernel of this surjection Π Σ ν I S log may be identiﬁed with the proﬁnite group gp Σ , and one veriﬁes immediately [from the deﬁnition Hom(M e gp /M S ⊗ Q/Z, k × ) ⊗ Z of G] that this kernel maps isomorphically onto Π e ⊆ Π I . In particular, it follows that we obtain an injection Π Σ ν → Π I whose image contains Π e and surjects onto I. Σ it follows that the image Im(Π Σ Since Π ν is abelian, ν ) of this injection is contained in I e ; since I e Π G = Π e [cf. [Mzk13], Proposition 1.2, (ii)], we thus conclude that Im(Π Σ ν ) = I e . Now it follows immediately from the deﬁnitions that I v , I v ⊆ I e ; moreover, one computes immediately that [in the notation of Example 1.1, (iii)] the Σ subgroups I v , I v ⊆ I e correspond to the subgroups of Hom(M e gp ⊗ Q/Z, k × ) ⊗ Z consisting of homomorphisms that vanish on ξ, η, respectively. Now the various assertions contained in the statement of assertion (ii) follow immediately. This completes the proof of assertion (ii). Next, we consider assertion (iii). Since Π v is slim [cf., e.g., [Mzk13], Remark 1.1.3] and commensurably terminal in Π G [cf. [Mzk13], Proposition 1.2, (ii)], it follows that D v Π G = Π v and I v Π G = {1}, so we obtain a natural injection I v → I. The fact that this injection is, in fact, surjective is immediate from the deﬁnitions when X is smooth over k and follows from the computation of “I v ” performed in the proof of assertion (ii) when X is singular. Next, let us observe [by deﬁnition!] with Π v , we obtain a natural morphism that since I v commutes I v × Π v → D v Π I , which is both injective [since I v Π v = {1}] and surjective ∼ [cf. the isomorphism I v → I; the fact that D v Π G = Π v ]. Now suppose that e is a cusp that abuts to v. Then [for appropriate choices of conjugates] it follows Π , I ⊆ D , immediately from the deﬁnitions that we have inclusions I e v e I and that I e commutes with I v . Note, moreover, that D e Π G = I e [cf. [Mzk13], Proposition 1.2, (ii)]. Thus, the fact that the natural projection I v → I is an isomorphism implies that we have a natural exact sequence 1 → I e → D e Π I → I → 1, and that the natural morphism I e × I v → D e Π I is an isomorphism. This completes the proof of assertion (iii). Next, we consider assertion (vii). Since D e Π G = Π e [cf. [Mzk13], Proposi- tion 1.2, (ii)], it follows that D e ⊆ C Π H (D e ) ⊆ C Π H (Π e ); on the other hand, by [Mzk13], Proposition 1.2, (i), it follows that C Π H (Π e ) = N Π H (Π e ) (= D e ); thus, D e = C Π H (D e ) = C Π H (Π e ) = N Π H (Π e ), as desired. Now it remains only to con- sider the case where e is a node. In this case, since I e is abelian [cf. assertion (ii)], it follows that I e ⊆ Z Π I (I e ) ⊆ C Π I (I e ) ⊆ C Π I (I e Π G ) = C Π I (Π e ); thus, the fact that D e Π I = C Π I (Π e ) = C Π I (I e ) = I e follows from the fact that I e surjects onto I [cf. assertion (ii)], together with the commensurable terminality of Π e in Π G [cf. [Mzk13], Proposition 1.2, (ii)]. This completes the proof of assertion (vii). Next, we consider assertion (iv). Suppose that (2) holds. Then it follows from assertions (ii), (iii) [and the deﬁnitions] that I v I v = {1}, I e = Π e · I v = 14 SHINICHI MOCHIZUKI Π e · I v ⊆ D v D v Π I ⊆ Z Π I (I v × I v ) ⊆ C Π I (I e ). On the other hand, by assertion (vii), we have C Π I (I e ) = I e . But this implies that D v D v Π I = I e . Next, suppose that (3) holds, and that Π v Π v = {1}. Then it follows immediately from our discussion of the situation in which (2) holds [cf. also assertion (iii)] that Σ ∼ Π I → I [so I v = D v D v Π I ], I v I v ⊆ I v I v = ( Z =) I v ⊆ D v D v I v I v = {1}. Thus, to complete the proof of assertion (iv), it suﬃces to verify — under the assumption that (1) and (2) are false! — that we have an equality (D v D v Π I ⊇) Π v Π v = {1}, and that (3) holds. Write C v , C v for the irreducible components of X corresponding to v, v . Suppose that both (1) and (2) are false. Recall that Π G [cf. [MT], Remark 1.2.2] Σ ), hence also Π I , are torsion-free. Thus, I v × Π v = D v Π I [cf. and I ( ∼ = Z assertion (iii)] is torsion-free, so by replacing Π H by an open subgroup of Π H [cf. Remark 1.2.1], we may assume without loss of generality that G [i.e., X log ] is sturdy [cf. §0], and that G is edge-paired [cf. §0]. Also, by projecting to the maximal pro-l quotients, for some l ∈ Σ, of suitable open subgroups [cf. Remark 1.2.1] of the various pro-Σ groups involved, one veriﬁes immediately that we may assume without loss of generality [for the remainder of the proof of assertion (iv)] that Σ = {l}. When l = 2, we may also assume without loss of generality [by replacing Π H by an open subgroup of Π H ] that G is edge-even [cf. §0]. Now I claim that D v D v Π G = Π v Π v = {1}. Indeed, suppose that Π v Π v = {1}. Then one veriﬁes immediately that there exist log admissible coverings [cf. [Mzk1], §3] Y log → X log × S log T log , corresponding to open subgroups J ⊆ Π G , which are split over C v [so Π v ⊆ J, Π v Π v ⊆ Π v J], but determine arbitrarily small neighborhoods Π v J of the identity element in Π v . [Here, we note that the existence of such coverings follows immediately from the fact that X log is edge-paired for arbitrary l and edge-even when l = 2. That is to say, one starts by constructing the covering over C v in such a way that the ramiﬁcation indices at the nodes and cusps of C v are all equal; one then extends the covering over the irreducible components of X adjacent to v [by applying the fact that X log is edge-paired for arbitrary l and edge-even when l = 2 — cf. the discussion of §0] in such a way that the covering is unramiﬁed over the nodes of these irreducible components that do not abut to C v ; ﬁnally, one extends the covering to a split covering over the remaining portion of X [which includes C v !].] But the existence of such J implies that Π v Π v = {1}, a contradiction. This completes the proof of the claim. Thus, the natural projection D v D v Π I → I has nontrivial open image [since Σ = {l}], which we denote by I v,v ⊆ I. Moreover, to complete the proof of assertion (iv), it suﬃces to derive a contradiction under the assumption that (1), (2), and (3) are false. Thus, for the remainder of the proof of assertion (iv), we assume that (1), (2), and (3) are false. Write C v + ⊆ X for the union of C v and the irreducible components of X that are adjacent to C v . We shall refer to a vertex of G as a C v + -vertex if it corresponds to an irreducible component of C v + . We shall say that a node e is a bridge node if it abuts both to a C v + -vertex and to a non-C v + -vertex. Thus, no bridge node abuts to v. Now let us write i v for the least common multiple of the indices i e of the bridge nodes e; i Σ v for the largest nonnegative power of l dividing i v . Let def def Σ d = l · i v · [I : I v,v ]; d Σ = l · i Σ v · [I : I v,v ] [so d is the largest positive power of l dividing d]. Here, we observe that for any open subgroup J 0 ⊆ Π I such that TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II D v J 0 surjects onto I [cf. assertion (iii)], and D v [I : I v,v ] = [(D v J 0 ) : (Π v J 0 ) · (D v 15 Π I ⊆ J 0 , it holds that D v Π I )] J 0 ]. Thus, it suﬃces to construct [where we note that D v D v Π I ⊆ D v onto open subgroups J ⊆ J 0 ⊆ Π I such that D v J 0 surjects I, Π v J 0 ⊆ J, Π I Π I ) ⊆ D v J], but ⊆ J [which implies that (Π v J 0 ) · (D v D v and D v [D v J 0 : D v J] > [I : I v,v ]. To this end, let us ﬁrst observe that the characteristic sheaf of the log scheme X log admits a section ζ over X satisfying the following properties: (a) ζ vanishes on the open subscheme of X given by the complement of C v + [hence, in particular, on C v ]; (b) ζ coincides with i v · σ ∈ M S [cf. the notation of Example 1.1, (iii)] at the generic points of C v + ; (c) ζ coincides with either (i v /i e )·ξ ∈ M e or (i v /i e )·η ∈ M e [cf. the notation of Example 1.1, (iii)] at each bridge node e. [Indeed, the existence of such a section ζ follows immediately from the discussion of Example 1.1, (iii), together with our deﬁnition of i v .] Thus, by taking the inverse image of ζ in the monoid that deﬁnes the log structure of X log , we obtain a line bundle L on X. Let Y → X be a ﬁnite étale cyclic covering of order a positive power of l such that L| Y has degree divisible by d Σ on every irreducible component of Y , and Y → X restricts to a connected covering over every irreducible component def def of X that is = C v [e.g., C v ], but splits over C v ; Y log = X log × X Y ; C w = C v × X Y ; def + = C v + × X Y . [Note that the fact that such a covering exists follows immediately C w from our assumption that G is sturdy.] Now let Z log → Y log be a log étale cyclic covering of degree d Σ satisfying the following properties: (d) Z log → Y log restricts to an étale covering of schemes over the complement + and splits over the irreducible components of Y that lie over C v of C w [cf. (a); the fact that (1), (2), and (3) are assumed to be false!]; (e) Z log → Y log is ramiﬁed, with ramiﬁcation index d Σ /i Σ v , over the generic + , but induces the trivial extension of the function ﬁeld of C w points of C w [cf. (b)]; (f) for each node f of Y that lies over a bridge node e of X, the restriction of Z log → Y log to the branch of f that does not abut to an irreducible + Σ is ramiﬁed, with ramiﬁcation index d Σ · i Σ component of C w e /i v [cf. (c)]. Indeed, to construct such a covering Z log → Y log , it suﬃces to construct a covering + [which is always possible, by the satisfying (d), (f) over the complement of C w conditions imposed on Y , together with the fact that (1), (2), and (3) are assumed 16 SHINICHI MOCHIZUKI to be false], and then to glue this covering to a suitable [i.e., such that (e) is def + log + ) = Y log × Y C w [by an fs log satisﬁed!] Kummer log étale covering of (C w Σ scheme!] obtained by extracting a d -th root of L| C w + [cf. the divisibility condition + on the degrees of L over the irreducible components of C w ]. [Here, we regard the G m -torsor determined by L| C w + as a subsheaf of the monoid deﬁning the log structure + log ) .] Now if we write J Z ⊆ J Y ⊆ Π I for the open subgroups deﬁned by the of (C w log log log onto I; D v Π I ⊆ J Y . On coverings Z → Y → X , then D v J Y surjects the other hand, Π v J Y ⊆ J Z [cf. (e)] and D v Π I ⊆ J Z [cf. (d)], while [cf. (e)] [D v J Y : D v J Z ] = d Σ /i Σ v > [I : I v,v ] def def [since d Σ = l · i Σ v · [I : I v,v ]]. Thus, it suﬃces to take J 0 = J Y , J = J Z . This completes the proof of assertion (iv). Next, we consider assertion (v). First, let us observe that it follows from assertion (iv) [i.e., by applying assertion (iv) to various open subgroups of Π H , Π I (γ · I v · γ −1 ) = {1}, then Π v = — cf. also Remark 1.2.1] that if, for γ ∈ Π H , I v γ · Π v · γ −1 . Thus, we conclude that N Π H (I v ) ⊆ C Π H (I v ) ⊆ N Π H (Π v ) = D v . On the other hand, since [by deﬁnition] I v = Z Π I (Π v ), and I is normal in H, it follows (Π v ) ⊆ N Π H (I v ), so N Π H (I v ) = C Π H (I v ) = D v , as desired. In that D v = N Π H particular, D v Π I = N Π I (I v ) = C Π I (I v ). Next, let us observe that D v Π G = Π v [cf. [Mzk13], Proposition 1.2, (ii)]. Thus, D v ⊆ C Π H (D v ) ⊆ C Π H (Π v ). Moreover, by [Mzk13], Proposition 1.2, (i), it follows that C Π H (Π v ) = N Π H (Π v ) = D v ; thus, terminal in we conclude that D v (respectively, D v Π I ; D v Π G ) is commensurably assertion (iii), we have D v Π I = I v × Π v ⊆ Π H (respectively, Π I ; Π G ). Finally, by Z Π I (I v ) ⊆ N Π I (I v ) = D v Π I , so D v Π I = Z Π I (I v ), as desired. This completes the proof of assertion (v). Next, we consider assertion (vi). The fact that the image of D v in H is open follows immediately from the fact that since the semi-graph G is ﬁnite, some open subgroup of H necessarily ﬁxes v. On the other hand, if G admits a vertex v = v, then Π v Π v is not open in Π v [cf. [Mzk13], Proposition 1.2, (i)]; since D v Π G = Π v [cf. [Mzk13], Proposition 1.2, (ii)], this implies that D v is not open in Π H . This completes the proof of assertion (vi). Next, we consider assertion (viii). First, we observe that if property (2) holds, Σ ∼ then (ii), (iii), [for appropriate choices of conjugates] ( Z =) I v ⊆ by assertions D e D e Π I → I, so I v = D e D e Π I . Thus, it suﬃces to verify that ei- ther (1) or (2) holds. Next, let us observe that, since, as observed above, Π I , hence also D e D e Π I , is torsion-free, by projecting to the maximal pro-l quo- tients, for some l ∈ Σ, of suitable open subgroups [cf. Remark 1.2.1] of the var- ious pro-Σ groups involved, one veriﬁes immediately we may assume without loss of generality [for the remainder of the proof of assertion (viii)] that Σ = {l}. Now if D e D e Π G = {1}, then [since Σ = {l}] D e D e Π G is open in Σ ), D e Π G ( ∼ Σ ) [cf. assertions (ii), (iii), (vii)], so we con- D e Π G ( ∼ = Z = Z clude from [Mzk13], Proposition 1.2, (i), that e = e . Thus, to complete the proof of assertion (viii), it suﬃces to derive a contradiction under the further assump- tion that D e D e Π G = {1}, and e and e do not abut to a common vertex. Moreover, by replacing Π H by an open subgroup of Π H [cf. Remark 1.2.1], we may TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 17 assume without loss of generality that G [i.e., X log ] is sturdy [cf. §0], and that G is edge-paired [cf. §0] for arbitrary l and edge-even [cf. §0] when l = 2. Now if, say, e is a cusp that abuts to a vertex v, then one veriﬁes immediately that there exist log étale cyclic coverings Y log → X log of degree an arbitrarily large power of l which are totally ramiﬁed over e, but unramiﬁed over the nodes of X, as well as over the cusps of X that abut to vertices = v. [Indeed, the existence of such coverings follows immediately from the fact that X log is edge-paired for arbitrary l and edge-even when l = 2 — cf. the discussion of §0.] In particular, such coverings are unramiﬁed over e , as well as over the generic point of the irreducible component of X corresponding to correspond v, hence to open subgroups J ⊆ Π I such that D e Π I ⊆ J D D e Π I ⊆ J [so D e e Π I ], and, moreover, J may be chosen so that the subgroup J D e Π I ⊆ D e Π I = I e × I v [cf. assertion (iii)] forms an arbitrarily small neighborhood of I v . Thus, we conclude that D e D e Π I ⊆ I v . vertex v , then [by symmetry] On the other hand, if e is also a cusp that abuts to a we conclude that D e D e Π I ⊆ I v , hence that I v I v = {1}. But, by assertion (iv), this implies that v = v , a contradiction. Thus, we may assume that, say, e is a node, so I e = D e Π I [cf. assertion (vii)]. Write v 1 , v 2 for the two distinct vertices to which e abuts; C 1 , C 2 for the def irreducible components of X corresponding to v 1 , v 2 ; C = C 1 C 2 ⊆ X; U C ⊆ C for the open subscheme obtained by removing the nodes that abut to vertices = v 1 , v 2 . Let us refer to the nodes and cusps of U C as inner, to the nodes of X that were removed from C to obtain U C as bridge nodes, and to the nodes and cusps of X which are neither inner nodes/cusps nor bridge nodes as external. [Thus, e is inner; e is external.] Observe that the natural projection to I yields an inclusion D e D e Π I → I with open image [since Σ = {l}]; denote the image of this inclusion by I C . Write i C for the least common multiple of the indices i f of the bridge nodes f ; i Σ C for the largest nonnegative power of l dividing i C . Let def def Σ d = l · i C · [I : I C ]; d Σ = l · i Σ C · [I : I C ] [so d is the largest positive power of l dividing d]. Here, we observe that [I : I C ] = [I e : Π e · (D e D e Π I )] [cf. assertion (ii)]. subgroup Then it suﬃces to construct an open J ⊆ Π I such that Π e ⊆ J and D e Π I ⊆ J [which implies that Π e · (D e D e Π I ) ⊆ I e J], but [I e : I e J] > [I : I C ]. X log To this end, let us ﬁrst observe that the characteristic sheaf of the log scheme admits a section ζ over X satisfying the following properties: (a) ζ vanishes on the open subscheme of X given by the complement of C; (b) ζ coincides with i C · σ ∈ M S [cf. the notation of Example 1.1, (iii)] at def the generic points of C = C 1 C 2 ; (c) ζ coincides with either (i C /i f ) · ξ ∈ M f or (i C /i f ) · η ∈ M f [cf. the notation of Example 1.1, (iii), where we take “e” to be f ] at each bridge node f . 18 SHINICHI MOCHIZUKI [Indeed, the existence of such a section ζ follows immediately from the discussion of Example 1.1, (iii), together with our deﬁnition of i C .] Thus, by taking the inverse image of ζ in the monoid that deﬁnes the log structure of X log , we obtain a line bundle L on X. Let Y → X be a ﬁnite étale Galois covering of order a positive power of l such that L| Y has degree divisible by d Σ on every irreducible component of Y , and Y → X restricts to a connected covering over every irreducible def component of X; Y log = X log × X Y . [Note that the fact that such a covering exists follows immediately from our assumption that G is sturdy.] Write C 1 Y , C 2 Y for the irreducible components of Y lying over C 1 , C 2 , respectively; we shall apply the terms “internal”, “external”, and “bridge” to nodes/cusps of Y that lie over such nodes/cusps of X. Now let Z log → Y log be a log étale cyclic covering of degree d Σ satisfying the following properties: (d) Z log → Y log restricts to an étale covering of schemes over the comple- def ment of C Y = C 1 Y C 2 Y [cf. (a)], hence, in particular, over the external nodes/cusps of Y ; (e) Z log → Y log is ramiﬁed, with ramiﬁcation index d Σ /i Σ C , over the generic Y Y points of C 1 , C 2 , and, at each internal node of Y lying over e, determines a covering corresponding to an open subgroup of I e that contains Π e [cf. (b)]; (f) for each bridge node f of Y , the restriction of Z log → Y log to the branch Σ of f that does not abut to C Y is ramiﬁed, with ramiﬁcation index d Σ ·i Σ f /i C [cf. (c)]. Indeed, to construct such a covering Z log → Y log , it suﬃces to construct a covering satisfying (d), (f) over the complement of C Y [which is always possible, by the conditions imposed on Y ], and then to glue this covering to a suitable [i.e., such def that (e) is satisﬁed!] Kummer log étale covering of (C Y ) log = Y log × Y C Y [by an fs log scheme!] obtained by extracting a d Σ -th root of L| C Y [cf. the divisibility condition on the degrees of L| C 1 Y , L| C 2 Y ]. [Here, we regard the G m -torsor determined by L| C Y as a subsheaf of the monoid deﬁning the log structure of (C Y ) log .] Now if the open subgroups deﬁned by the coverings Z log → we write J Z ⊆ J Y ⊆ Π I for Y log → X log , then I e , D e Π I ⊆ J Y . On the other hand, Π e ⊆ J Z [cf. (e)] and D e Π I ⊆ J Z [cf. (d)], while [cf. (e)] [I e : I e J Z ] = d Σ /i Σ C > [I : I C ] def [since d Σ = l · i Σ C · [I : I C ]]. Thus, it suﬃces to take J = J Z . This completes the proof of assertion (viii). Next, we consider assertion (ix). The fact that the image of D e in H is open fol- lows immediately from the fact that since the semi-graph G is ﬁnite, some open sub- group of H necessarily ﬁxes e. On the other hand, since D e Π G = N Π G (Π e ) = Π e [cf. [Mzk13], Proposition 1.2, (ii)] is abelian, hence not open in the slim, nontrivial TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 19 proﬁnite group Π G , it follows that D e is not open in Π H . This completes the proof of assertion (ix). Finally, we consider assertion (x). First, let us observe that an easy com- putation reveals that if the image of τ S is a non-nodal, non-cuspidal point of the irreducible component of X corresponding to a vertex v τ of G, then τ I (I) = I v τ . Next, let us suppose that the image of τ S is the node of X corresponding to some node e τ of G. Then an easy computation [cf. the computations performed in the proof of assertion (ii)] reveals that τ I (I) ⊆ I e τ , but that τ I (I) is not contained in I v for any vertex v to which e τ abuts. If, moreover, τ I (I) ⊆ I v for some vertex v to which e τ does not abut, then [since the very existence of the node e τ implies that X is singular] there exists a node e = e τ that abuts to v, so τ I (I) ⊆ I v ⊆ I e [cf. as- sertion (ii)]; but this implies that τ I (I) ⊆ I e I e τ , so, by assertion (viii), it follows that τ I (I) ⊆ I v for some vertex to which both e and e τ abut — a contradiction. Thus, in summary, we conclude that in this case, τ I is non-verticial. Now suppose that τ I is non-verticial and non-edge-like. Then the observations of the preceding paragraph imply that the image of τ S is a cusp of X. Write e τ for the corresponding cusp of G. Thus, one veriﬁes immediately that τ I (I) ⊆ D e τ . The uniqueness of e τ then follows from assertion (viii) [and the fact that τ I is non- verticial]. Thus, for the remainder of the proof of assertion (x), we may assume that the image of τ S is not a cusp. Now the remainder of assertion (x) follows formally, in light of what of we have done so far, from assertions (iv), (viii). This completes the proof of assertion (x). Corollary 1.4. (Graphicity of Isomorphisms of (D)PSC-Extensions) Let l be a prime number. For i = 1, 2, let 1 → Π G i → Π H i → H i → 1 be an l-cyclotomically full [cf. Remark 1.2.3] DPSC-extension (respectively, PSC- extension), associated to construction data (X i log → S i log , Σ i , G i , ρ H i , ι i ) (re- spectively, (X i log → S i log , Σ i , G i , ρ H i )) such that l ∈ Σ i ; in the non-resp’d case, write I i ⊆ H i for the inertia subgroup. Let ∼ φ H : H 1 → H 2 ; ∼ φ Π : Π G 1 → Π G 2 be compatible [i.e., with the respective outer actions of H i on Π G i ] isomorphisms of proﬁnite groups; in the non-resp’d case, suppose further that φ H (I 1 ) = I 2 . Then Σ 1 = Σ 2 ; φ Π is graphic [cf. [Mzk13], Deﬁnition 1.4, (i)], i.e., arises from ∼ an isomorphism of semi-graphs of anabelioids G 1 → G 2 . Proof. This follows immediately from [Mzk13], Corollary 2.7, (i), (iii). Here, as in the proof of [Mzk13], Corollary 2.8, we ﬁrst apply [Mzk13], Corollary 2.7, (i) [which suﬃces to complete the proof of Corollary 1.4 in the resp’d case and allows one to reduce to the noncuspidal case in the non-resp’d case], then apply [Mzk13], Corollary 2.7, (iii), to the compactiﬁcations of corresponding sturdy ﬁnite étale coverings of the G i . We are now ready to deﬁne a purely group-theoretic, combinatorial analogue of the notion of a stable polycurve given in [Mzk2], Deﬁnition 4.5. 20 SHINICHI MOCHIZUKI Deﬁnition 1.5. We shall refer to an extension of proﬁnite groups as a PPSC- extension [i.e., “poly-PSC-extension”] if, for some positive integer n and some nonempty set of primes Σ, it admits a “structure of pro-Σ PPSC-extension of dimension n”. Here, for n a positive integer, Σ a nonempty set of primes, and 1 → Δ → Π → H → 1 an extension of proﬁnite groups, we deﬁne the notion of a structure of pro-Σ PPSC- extension of dimension n as follows [inductively on n]: (i) A structure of pro-Σ PPSC-extension of dimension 1 on the extension 1 → Δ → Π → H → 1 is deﬁned to be a structure of pro-Σ PSC-extension. Suppose that the extension 1 → Δ → Π → H → 1 is equipped with a structure of pro-Σ PPSC- extension of dimension 1. Thus, we have an associated semi-graph of anabelioids G, together with a continuous action of H on G, and a compatible isomorphism ∼ Δ → Π G . We deﬁne the [horizontal] divisors of this PPSC-extension to be the cusps of the PSC-extension 1 → Δ → Π → H → 1. Thus, each divisor c of the PPSC-extension 1 → Δ → Π → H → 1 has associated inertia and decomposition groups I c ⊆ D c ⊆ Π [cf. Deﬁnition 1.2, (i)]. Moreover, by [Mzk13], Proposition 1.2, (i), a divisor is completely determined by [the conjugacy class of] its inertia group, as well as by [the conjugacy class of] its decomposition group. Finally, we shall refer to the extension 1 → Δ → Π → H → 1 [itself] as the ﬁber extension associated to the PPSC-extension 1 → Δ → Π → H → 1 of dimension 1. (ii) A structure of pro-Σ PPSC-extension of dimension n + 1 on the extension 1 → Δ → Π → H → 1 is deﬁned to be a collection of data as follows: def (a) a quotient Π Π ∗ such that Δ † = Ker(Π Π ∗ ) ⊆ Δ; thus, the image Δ ∗ ⊆ Π ∗ of Δ in Π ∗ determines an extension 1 → Δ ∗ → Π ∗ → H → 1 — which we shall refer to as the associated base extension; the subgroup Δ † ⊆ Π determines an extension 1 → Δ † → Π → Π ∗ → 1 — which we shall refer to as the associated ﬁber extension; (b) a structure of pro-Σ PPSC-extension of dimension n on the base extension 1 → Δ ∗ → Π ∗ → H → 1; (c) a structure of pro-Σ PPSC-extension of dimension 1 on the ﬁber extension 1 → Δ † → Π → Π ∗ → 1; (d) for each base divisor [i.e., divisor of the base extension] c ∗ , a structure of DPSC-extension on the extension 1 → Δ † → Π c ∗ = Π × Π ∗ D c ∗ → D c ∗ → 1 def — which we shall refer to as the extension at c ∗ — which is compatible with the PSC-extension structure on the ﬁber extension [cf. (c)], in the TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 21 sense that both structures yield the same cuspidal inertia subgroups ⊆ Δ † ; also, we require that the inertia subgroup of D c ∗ [i.e., that arises from this structure of DPSC-extension] be equal to I c ∗ . In this situation, we shall refer to Σ as the ﬁber prime set of the PPSC-extension 1 → Δ → Π → H → 1; we shall refer to as a divisor of the PPSC-extension 1 → Δ → Π → H → 1 any element of the union of the set of cusps — which we shall refer to as horizontal divisors — of the PSC-extension 1 → Δ † → Π → Π ∗ → 1 and, for each base divisor c ∗ , the set of vertices of the DPSC-extension 1 → Δ † → Π c ∗ → D c ∗ → 1 — which we shall refer to as vertical divisors [lying over c ∗ ]. Thus, each divisor c of the PPSC-extension 1 → Δ → Π → H → 1 has associated inertia and decomposition groups I c ⊆ D c ⊆ Π. In particular, whenever c is vertical and lies over a base divisor c ∗ , we have I c ⊆ D c ⊆ Π c ∗ . Remark 1.5.1. Thus, [the collection of ﬁber extensions arising from] any struc- ture of PPSC-extension of dimension n on an extension 1 → Δ → Π → H → 1 determine two compatible sequences of surjections def def def def Δ n = Δ Δ n−1 . . . Δ 1 Δ 0 = {1} Π n = Π Π n−1 . . . Π 1 Π 0 = H such that each [extension determined by a] surjection Π m Π m−1 , for m = 1, . . . , n, is a ﬁber extension [hence equipped with a structure of PSC-extension]; Δ m = Ker(Π m H). If c = c n is a divisor of [the extension determined by] Π = Π n , then [cf. Deﬁnition 1.5, (ii)] there exists a uniquely determined sequence of divisors c n → c n−1 → . . . → c n c −1 → c n c — where n c ≤ n is a positive integer; for m = n c , . . . , n, c m is a divisor of Π m ; c n c is a horizontal divisor; the notation “ →” denotes the relation of “lying over” [so c m+1 is a vertical divisor that lies over c m , for n c ≤ m < n] — together with sequences of [conjugacies classes of ] inertia and decomposition groups I c n → I c n−1 → . . . → I c nc −1 → I c nc D c n → D c n−1 → . . . → D c nc −1 → D c nc [i.e., for n c ≤ m < n, I c m+1 ⊆ Π m+1 maps into I c m ⊆ Π m , and D c m+1 ⊆ Π m+1 maps into D c m ⊆ Π m ]. Remark 1.5.2. Let 1 → Δ → Π → H → 1 be a PPSC-extension of dimension n [where n is a positive integer]. Then one veriﬁes immediately that if Π • ⊆ Π is any open subgroup of Π, then there exists an open subgroup Π •• of Π • that [when equipped with the induced extension structure] admits a structure of PPSC- extension of dimension n — cf. Remark 1.2.1. Here, we note that one must, in general, pass to “some open subgroup” Π •• of Π • in order to achieve a situation in which all of the ﬁber [PSC-]extensions have “stable reduction” [cf. Remark 1.2.1; Deﬁnition 1.5, (ii), (d)]. 22 SHINICHI MOCHIZUKI Remark 1.5.3. For l a prime number, we shall say that a PPSC-extension 1 → Δ → Π → H → 1 of dimension n is l-cyclotomically full if each of its n associated ﬁber extensions [cf. Remark 1.5.1] is l-cyclotomically full as a PSC-extension [cf. Remark 1.2.3]. Thus, it follows immediately from the ﬁnal portion of Remark 1.2.3 that the issue of whether or not the PPSC-extension 1 → Δ → Π → H → 1 is l-cyclotomically full depends only on the sequence of surjections of proﬁnite groups Π n Π n−1 . . . Π 1 Π 0 [cf. Remark 1.5.1]. Remark 1.5.4. def Let k be a ﬁeld; k a separable closure of k; G k = Gal(k/k); def S = Spec(k); Z log → S the log scheme determined by a stable polycurve over S — i.e., Z log admits a successive ﬁbration by generically smooth stable log curves [cf. [Mzk2], Deﬁnition def 4.5, for more details]; U Z ⊆ Z the interior of Z log ; D Z = Z\U Z [with the reduced induced structure]; n the dimension of the scheme Z; def def 1 → Δ Z = π 1 (U Z × k k) → Π Z = π 1 (U Z ) → G k → 1 the exact sequence of étale fundamental groups [well-deﬁned up to inner automor- phism] associated to the structure morphism U Z → S. Then by repeated appli- cation of the discussion of Example 1.1 to the ﬁbers of the successive ﬁbration [mentioned above] of Z log by stable log curves, one veriﬁes immediately that: (i) If k is of characteristic zero, then the structure of stable polycurve on Z log determines a structure of proﬁnite PPSC-extension of dimension n on the extension 1 → Δ Z → Π Z → G k → 1. Moreover, one veriﬁes immediately that: (ii) In the situation of (i), the divisors of Π Z [in the sense of Deﬁnition 1.5] are in natural bijective correspondence with the irreducible divisors of D Z in a fashion that is compatible with the inertia and decomposition groups of divisors of Π Z [in the sense of Deﬁnition 1.5] and of irreducible divisors of D Z [in the usual sense]. Finally, whether or not k is of characteristic zero, depending on the structure of Z log [cf., e.g., Corollary 1.10 below], various quotients of the extension 1 → Δ Z → Π Z → G k → 1 may be equipped with a structure of pro-Σ PPSC-extension [induced by the structure of stable polycurve on Z], for various nonempty sets of prime numbers Σ that are not equal to the set of all prime numbers; a similar observation to (ii) concerning a natural bijective correspondence of “divisors” then applies to such quotients. When considering such quotients 1 → Δ → Π → H → 1 of the extension 1 → Δ Z → Π Z → G k → 1, it is useful to observe that the slimness of Δ [cf. Proposition 1.6, (i), below] implies that such a quotient Π Z Π is completely determined by the induced quotients Δ Z Δ, G k H [cf. the discussion of the out notation “ ” in §0]; we shall refer to such a quotient 1 → Δ → Π → H → 1 as TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 23 a PPSC-extension arising from Z log , k — where we write k ⊆ k for the subﬁeld ﬁxed by Ker(G k H). Proposition 1.6. (Basic Properties of PPSC-Extensions) Let 1 → Δ → Π → H → 1 be a pro-Σ PPSC-extension of dimension n [where n is a positive integer]; 1 → Δ † → Π → Π ∗ → 1 the associated ﬁber extension; c, c divisors of Π. Then: (i) Δ is slim. In particular, if H is slim, then so is Π. (ii) D c is commensurably terminal in Π. Σ . (iii) We have: C Π (I c ) = D c . As abstract proﬁnite groups, I c ∼ = Z (iv) D c is not open in Π. The divisor c is horizontal if and only if D c projects to an open subgroup of Π ∗ . If c is vertical and lies over a base divisor c ∗ , then D c projects onto an open subgroup of D c ∗ . (v) If D c D c is open in D c , D c , then c = c . In particular, a divisor of Π is completely determined by its associated decomposition group. (vi) If I c I c is open in I c , I c , then c = c . In particular, a divisor of Π is completely determined by its associated inertia group. Proof. Assertion (i) follows immediately from the “slimness of Π G ” discussed in Example 1.1, (ii) [cf. Deﬁnition 1.5, (i); Deﬁnition 1.5, (ii), (c)]. Next, we consider assertion (ii). We apply induction on n. If c is horizontal, then assertion (ii) follows from [the argument applied in] Proposition 1.3, (vii) [cf. also Deﬁnition 1.5, (ii), (c)]. If c is vertical, then c lies over some base divisor c ∗ , and we are in the situation of Deﬁnition 1.5, (ii), (d). By Proposition 1.3, (vi), it follows that D c surjects onto some open subgroup of D c ∗ , hence that C Π (D c ) maps into C Π ∗ (D c ∗ ); by the induction hypothesis, C Π ∗ (D c ∗ ) = D c ∗ , so C Π (D c ) ⊆ Π c ∗ . Thus, the fact that C Π (D c ) = D c follows from Proposition 1.3, (v). This completes the proof of assertion (ii). Next, we consider assertion (iii). Again we apply induction on n. If c is horizontal, then assertion (iii) follows from [the argument applied in] Proposition 1.3, (i), (vii). If c is vertical, then c lies over some base divisor c ∗ , and we are in the situation of Deﬁnition 1.5, (ii), (d). By Proposition 1.3, (iii) [cf. Deﬁnition 1.5, ∼ Σ . In particular, C Π (I c ) maps into (ii), (d)], we have isomorphisms I c → I c ∗ ∼ = Z C Π ∗ (I c ∗ ); by the induction hypothesis, C Π ∗ (I c ∗ ) = D c ∗ . Thus, C Π (I c ) ⊆ Π c ∗ , so the fact that C Π (I c ) = D c follows from Proposition 1.3, (v). This completes the proof of assertion (iii). Next, we consider assertion (iv). Again we apply induction on n. If c is horizontal, then by [the argument applied in] Proposition 1.3, (ix), D c is not open in Π, but D c projects to an open subgroup of Π ∗ . If c is vertical, then c lies over some base divisor c ∗ , and we are in the situation of Deﬁnition 1.5, (ii), (d); D c ⊆ Π c ∗ . By Proposition 1.3, (vi), D c projects onto an open subgroup of D c ∗ . By the induction 24 SHINICHI MOCHIZUKI hypothesis, D c ∗ is not open in Π ∗ , so Π c ∗ is not open in Π; thus, D c is not open in Π, and its image in Π ∗ is not open in Π ∗ . This completes the proof of assertion (iv). Next, we consider assertion (v). Again we apply induction on n. By assertion (iv), c is horizontal if and only if c is. If c, c are horizontal, then the fact that c = c follows from [Mzk13], Proposition 1.2, (i), (ii). Thus, we may suppose that ∗ ∗ c, c are vertical and lie over respective base divisors c , (c ) . By assertion (iv), it follows that D c ∗ D (c ) ∗ is open in D c ∗ , D (c ) ∗ ; by the induction hypothesis, this implies that c ∗ = (c ) ∗ . Thus, by intersecting with “Π G ” [cf. Proposition 1.3, (v)] and applying [Mzk13], Proposition 1.2, (i), we conclude that c = c . This completes the proof of assertion (v). Finally, we observe that assertion (vi) is an immediate consequence of assertions (iii), (v). We are now ready to state and prove the main result of the present §1. Theorem 1.7. (Graphicity of Isomorphisms of PPSC-Extensions) Let l be a prime number; n a positive integer. For = α, β, let Σ be a nonempty set of primes; 1 → Δ → Π → H → 1 an l-cyclotomically full [cf. Remark 1.5.3] pro-Σ PPSC-extension of dimension n; Π n = Π Π n−1 . . . Π 1 Π 0 = H def def the sequence of successive ﬁber extensions associated to Π [cf. Remark 1.5.1]. Let ∼ φ : Π α → Π β ∼ β be an isomorphism of proﬁnite groups that induces isomorphisms φ m : Π α m → Π m , for m = 0, 1, . . . , n [so φ = φ n ]. Then: (i) We have Σ α = Σ β . (ii) For m ∈ {1, . . . , n}, φ m induces a bijection between the set of divisors α of Π m and the set of divisors of Π βm . β (iii) For m ∈ {1, . . . , n}, suppose that c α , c β are divisors of Π α m , Π m , respec- tively, that correspond via the bijection of (ii). Then φ m (I c α ) = I c β , φ m (D c α ) = D c β . That is to say, φ m is compatible with the inertia and decomposition groups of divisors. (iv) For m ∈ {0, . . . , n − 1}, the isomorphism ∼ β α β Ker(Π α m+1 Π m ) → Ker(Π m+1 Π m ) induced by φ m+1 is graphic [i.e., compatible with the semi-graphs of anabelioids that appear in the respective collections of construction data of the PSC-extensions Π m+1 Π m , for = α, β]. α (v) For m ∈ {1, . . . , n − 1}, c α , c β corresponding divisors of Π α m , Π m , the isomorphism ∼ β Ker((Π α m+1 ) c α D c α ) → Ker((Π m+1 ) c β D c β ) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 25 induced by φ m+1 is graphic [i.e., compatible with the semi-graphs of anabelioids that appear in the respective collections of construction data of the DPSC-extensions (Π m+1 ) c D c , for = α, β]. Proof. All of the assertions of Theorem 1.7 follow immediately from [the various deﬁnitions involved, together with] repeated application of Corollary 1.4 to the PSC- extensions Π m+1 Π m [cf. Deﬁnition 1.5, (ii), (c)] and the DPSC-extensions (Π m+1 ) c D c [cf. Deﬁnition 1.5, (ii), (d)], for = α, β. Remark 1.7.1. In Theorem 1.7, instead of phrasing the result as an asser- tion concerning the preservation of structures via some isomorphism between two PPSC-extensions, one may instead phrase the result as an assertion concerning the existence of an explicit “group-theoretic algorithm” for reconstructing, from a single given PPSC-extension, the various structures corresponding to graphicity, divisors, and inertia and decomposition groups of divisors — i.e., in the fashion of [Mzk15], Lemma 4.5, for cuspidal decomposition groups; a similar remark may be made concerning Corollary 1.4. [We leave the routine details to the interested reader.] Indeed, both Corollary 1.4 and Theorem 1.7 are, in essence, formal consequences of the “graphicity theory” of [Mzk13], which [just as in the case of [Mzk15], Lemma 4.5] consists precisely of such explicit “group-theoretic algorithms” for reconstruct- ing the various structures corresponding to graphicity in the case of semi-graphs of anabelioids of PSC-type. Before proceeding, we observe the following result, which is, in essence, inde- pendent of the theory of the present §1. Theorem 1.8. (PPSC-Extensions over Galois Groups of Arithmetic k a solvably Fields) For = α, β, let k be a ﬁeld of characteristic zero; def k /k ); closed [cf. [Mzk15], Deﬁnition 1.4] Galois extension of k ; H = Gal( (Z ) log the log scheme determined by a stable polycurve over k ; Σ a nonempty set of primes; n a positive integer; 1 → Δ → Π H → 1 a pro-Σ PPSC- k [cf. Remark 1.5.4]; extension of dimension n associated to (Z ) log , Π n = Π Π n−1 . . . Π 1 Π 0 = H def def the sequence of successive ﬁber extensions associated to Π [cf. Remark 1.5.1]. Let ∼ φ : Π α → Π β ∼ β be an isomorphism of proﬁnite groups that induces isomorphisms φ m : Π α m → Π m , for m = 0, 1, . . . , n [so φ = φ n ]. Then: (i) (Relative Version of the Grothendieck Conjecture for Stable Poly- curves over Generalized Sub-p-adic Fields) Suppose that for = α, β, k is generalized sub-p-adic [cf. [Mzk4], Deﬁnition 4.11] for some prime number ∼ p ∈ Σ α Σ β , and that the isomorphism of Galois groups φ 0 : H α → H β arises ∼ ∼ from a pair of isomorphisms of ﬁelds k α → k β , k α → k β . Then Σ α = Σ β ; there 26 SHINICHI MOCHIZUKI ∼ exists a unique isomorphism of log schemes (Z α ) log → (Z β ) log that gives rise to φ. (ii) (Absolute Version of the Grothendieck Conjecture for Stable Polycurves over Number Fields) Suppose that for = α, β, k is a number ﬁeld. Then Σ α = Σ β ; there exists a unique isomorphism of log schemes ∼ (Z α ) log → (Z β ) log that gives rise to φ. Proof. Assertions (i), (ii) follow immediately from repeated application of [Mzk4], Theorem 4.12 [cf. also [Mzk2], Corollary 7.4], together with [in the case of assertion (ii)] “Uchida’s theorem” [cf., e.g., [Mzk10], Theorem 3.1]. Finally, we study the consequences of the theory of the present §1 in the case of conﬁguration spaces. We refer to [MT] for more details on the theory of conﬁg- uration spaces. Deﬁnition 1.9. Let l be a prime number; Σ a set of primes which is either of cardinality one or equal to the set of all primes; X a hyperbolic curve of type (g, r) over a ﬁeld k of characteristic ∈ Σ; k a separable closure of k; n ≥ 1 an integer; X n the n-th conﬁguration space associated to X [cf. [MT], Deﬁnition 2.1, (i)]; E the index set [i.e., the set of factors — cf. [MT], Deﬁnition 2.1, (i)] of X n ; π 1 (X n × k k) Θ the maximal pro-Σ quotient of π 1 (X n × k k); Δ ⊆ Θ a product-theoretic open subgroup [cf. [MT], Deﬁnition 2.3, (ii)]; 1 → Δ → Π → H → 1 an extension of proﬁnite groups. ∼ (i) We shall refer to as a labeling on E a bijection Λ : {1, 2, . . . , n} → E. Thus, for each labeling Λ on E, we obtain a structure of hyperbolic polycurve [i.e., a collection of data exhibiting X n as a hyperbolic polycurve — cf. [Mzk2], Deﬁnition 4.6] on X n , arising from the various natural projection morphisms associated to X n [cf. [MT], Deﬁnition 2.1, (ii)], by projecting in the order speciﬁed by Λ. In particular, for each labeling Λ on E, we obtain a structure of PPSC-extension on [the extension 1 → Δ Λ → Δ Λ → {1} → 1 associated to] some open subgroup Δ Λ ⊆ Δ [which may be taken to be arbitrarily small — cf. Remark 1.5.2]. (ii) Let Λ be a labeling on E. Then we shall refer to a structure of PPSC- extension on [the extension 1 → Δ Λ → Π Λ → H Λ → 1 arising from 1 → Δ → Π → H → 1 by intersecting with] an open subgroup Π Λ ⊆ Π as Λ-admissible if it induces the structure of PPSC-extension on Δ Λ discussed in (i). (iii) We shall refer to as a structure of [pro-Σ] CPSC-extension [of type (g, r) and dimension n, with index set E] [i.e., “conﬁguration (space) pointed stable curve extension”] on the extension 1 → Δ → Π → H → 1 any collection of data as follows: for each labeling Λ on E, a Λ-admissible structure of PPSC-extension on some open subgroup Π Λ ⊆ Π [which may be taken to be arbitrarily small — cf. Remark 1.5.2]. We shall refer to a structure of CPSC-extension on Π as l-cyclotomically full if, for each labeling Λ on E, the Λ-admissible structure of PPSC-extension that TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 27 constitutes the given structure of CPSC-extension is l-cyclotomically full. We shall refer to a structure of CPSC-extension on Π as strict if one may take Δ to be Θ, and, for each labeling Λ on E, one may take Π Λ to be Π. We shall refer to 1 → Δ → Π → H → 1 as a [pro-Σ] CPSC-extension [of type (g, r) and dimension n, with index set E] if it admits a structure of CPSC-extension; if this structure of CPSC-extension may be taken to be l-cyclotomically full (respectively, strict), then we shall refer to the CPSC-extension itself as l-cyclotomically full (respectively, strict). If 1 → Δ → Π → H → 1 is a CPSC-extension, then we shall refer to (Σ, X, k, Θ) as construction data for this CPSC-extension. (iv) Let k ⊆ k be a solvably closed [cf. [Mzk15], Deﬁnition 1.4] Galois extension of k; suppose that Z → X n is a ﬁnite étale covering such that Z × k k → X n × k k is the [connected] covering determined by the open subgroup Δ ⊆ Θ [so we have a natural surjection π 1 (Z × k k) Δ]. Then [cf. the discussion of Remark 1.5.4] we shall refer to a [structure of] CPSC-extension [on] 1 → Δ → Π → H → 1 as arising from Z, k/k if there exist ∼ a surjection π 1 (Z) Π and an isomorphism Gal( k/k) → H that are compatible with one another as well as with the natural surjections π 1 (Z × k k) Δ, π 1 (Z) Gal(k/k) Gal( k/k) and, moreover, satisfy the property that the structure of CPSC-extension on 1 → Δ → Π → H → 1 is induced by the various structures of hyperbolic polycurve on Z, X n , associated to a suitable labeling of E [cf. (i)]. Corollary 1.10. (Cominatorial Conﬁguration Spaces) Let l be a prime number. For = α, β, let 1 → Δ → Π → H → 1 be an extension of proﬁnite groups equipped with some [ﬁxed!] l-cyclotomically full structure of CPSC-extension of type (g , r ) ∈ / {(0, 3), (1, 1)} and dimen- sion n , with index set E . If this ﬁxed structure of CPSC-extension is not strict for either = α or = β, then we assume that both g α , g β are ≥ 2. Let ∼ φ : Π α → Π β be an isomorphism of proﬁnite groups such that φ(Δ α ) = Δ β . Then: ∼ (i) The isomorphism φ determines a bijection E α → E β of index sets. In def particular, n α = n β , so we write n = n α = n β . (ii) For each pair of compatible [i.e., relative to the bijection of (i)] labelings Λ = (Λ α , Λ β ) of E α , E β , there exist open subgroups Π [for = α, β] Λ ⊆ Π β α such that the following properties hold: (a) φ(Π Λ ) = Π Λ ; (b) for = α, β, the open subgroup Π Λ admits an Λ -admissible structure of PPSC-extension; (c) if we write (Π Λ ) n = (Π Λ ) (Π Λ ) n−1 . . . (Π Λ ) 1 (Π Λ ) 0 = H Λ def def 28 SHINICHI MOCHIZUKI for the sequence of successive ﬁber extensions associated to the structures of PPSC-extension of (b) [cf. Remark 1.5.1], then φ induces isomorphisms ∼ β (Π α Λ ) m → (Π Λ ) m [for m = 0, . . . , n]. In particular, φ satisﬁes the hypotheses of Theorem 1.7. Proof. By [MT], Corollaries 4.8, 6.3 [cf. our hypotheses on (g , r )], φ induces ∼ a bijection E α → E β between the respective index sets, together with compatible isomorphisms between the various ﬁber subgroups of Δ α , Δ β . [Note that even though these results of [MT] are stated only in the case where the ﬁeld appearing in the construction data is of characteristic zero, the results generalize immediately to the case where this ﬁeld is of characteristic invertible in Σ , since any hyperbolic curve in positive characteristic may be lifted to a hyperbolic curve in characteristic zero in a fashion that is compatible with the maximal pro-Σ quotients of the étale fundamental groups of the associated conﬁguration spaces — cf., e.g., [MT], Proposition 2.2, (v).] To obtain open subgroups Π satisfying the desired Λ ⊆ Π properties, it suﬃces to argue by induction on n, by applying Remark 1.5.2. Remark 1.10.1. 1.10. A similar remark to Remark 1.7.1 may be made for Corollary Corollary 1.11. (Conﬁguration Spaces over Arithmetic Fields) For = k a solvably closed [cf. [Mzk15], Deﬁnition 1.4] α, β, let k be a perfect ﬁeld; / {(0, 3), (1, 1)} Galois extension of k ; X a hyperbolic curve of type (g , r ) ∈ over k ; n a positive integer; Z → (X ) n a geometrically connected [over k ] ﬁnite étale covering of the n -th conﬁgu- ration space (X ) n of X ; Σ a nonempty set of primes; 1 → Δ → Π → H → 1 an extension of proﬁnite groups equipped with some [ﬁxed!] structure of k [cf. Deﬁnition 1.9, (iv)]. If this pro-Σ CPSC-extension arising from Z , ﬁxed structure of CPSC-extension is not strict for either = α or = β, then we assume that both g α , g β are ≥ 2. Let ∼ φ : Π α → Π β be an isomorphism of proﬁnite groups. Then: (i) (Relative Version of the Grothendieck Conjecture for Conﬁgura- tion Spaces over Generalized Sub-p-adic Fields) Suppose, for = α, β, that k is generalized sub-p-adic [cf. [Mzk4], Deﬁnition 4.11] for some prime number ∼ p ∈ Σ α Σ β , and that φ lies over an isomorphism of Galois groups φ 0 : H α → H β ∼ ∼ that arises from a pair of isomorphisms of ﬁelds k α → k β , k α → k β . Then TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 29 ∼ Σ α = Σ β ; there exists a unique isomorphism of schemes Z α → Z β that gives rise to φ. (ii) (Strict Semi-absoluteness) Suppose, for = α, β, that k is either an FF, an MLF, or an NF [cf. [Mzk15], §0]. Then φ(Δ α ) = Δ β [i.e., φ is “strictly semi-absolute”]. (iii) (Absolute Version of the Grothendieck Conjecture for Conﬁgu- ration Spaces over MLF’s) Suppose, for = α, β, that k is an MLF, that n ≥ 2, that n ≥ 3 if X is proper, and that Σ is the set of all primes. Then ∼ there exists a unique isomorphism of schemes Z α → Z β that gives rise to φ. (iv) (Absolute Version of the Grothendieck Conjecture for Conﬁg- uration Spaces over NF’s) Suppose, for = α, β, that k is an NF. Then ∼ Σ α = Σ β ; there exists a unique isomorphism of schemes Z α → Z β that gives rise to φ. Proof. Assertion (i) (respectively, (iv)) follows immediately from Corollary 1.10, (ii), and Theorem 1.8, (i) (respectively, Theorem 1.8, (ii)) [applied to the coverings of Z α , Z β determined by the open subgroups of Corollary 1.10, (ii)]. Assertion (ii) follows immediately from [Mzk15], Corollary 2.8, (ii). Note that in the situation of assertion (ii), assertion (ii) implies that Σ α = Σ β [since Σ may be characterized as the unique minimal set of primes Σ such that Δ is a pro-Σ group]; moreover, in light of our assumptions β on α k , it β follows immediately that Π is l-cyclotomically α Σ =Σ =Σ . full for any l ∈ Σ Finally, we consider assertion (iii). First, let us observe that by Corollary 1.10, (ii), and Theorem 1.8, (i) [applied to the coverings of Z α , Z β determined by the open subgroups of Corollary 1.10, (ii)], it suﬃces to verify that the isomorphism ∼ φ H : H α → H β induced by φ [cf. assertion (ii)] arises from an isomorphism of ﬁelds ∼ k α → k β . To this end, let us observe that by Corollary 1.10, (ii), we may apply Theorem 1.7 to the present situation. Also, by Corollary 1.10, (i), n = n α = n β is always ≥ 2; moreover, if either of the X is proper, then n ≥ 3. Next, let us observe that if X is proper (respectively, aﬃne), then the stable log curve that appears in the logarithmic compactiﬁcation of the ﬁbration (X ) 3 → (X ) 2 (respectively, (X ) 2 → (X ) 1 ) over the generic point of the diagonal divisor of (X ) 2 (respectively, over any cusp of X ) contains an irreducible component whose interior is a tripod [i.e., a copy of the projective line minus three marked points]. In particular, if we apply Theorem 1.7, (iii), to the vertical divisor determined by such an irreducible component, then we may conclude that φ induces an isomorphism between the decomposition groups of these vertical divisors. In particular, [after possibly replacing the given k α , k β by corresponding ﬁnite extensions of k α , k β ] we obtain, for = α, β, a hyperbolic curve C over k , together with an isomorphism of proﬁnite groups ∼ k α ) → π 1 (C β × k β k β ) φ C : π 1 (C α × k α k ) correspond to the respective “Π v ’s” of the induced by φ [so the π 1 (C × k vertical divisors under consideration] that is compatible with the outer action of 30 SHINICHI MOCHIZUKI k ) and the isomorphism φ H ; moreover, here we may assume H on π 1 (C × k that, say, C α is a ﬁnite étale covering of a tripod. [In particular, we observe that k ) implies — cf. the the existence of the natural outer action of H on π 1 (C × k argument given in the proof of [Mzk10], Proposition 3.3 — that k is necessarily an algebraic closure of k .] On the other hand, since the “absolute p-adic version of the Grothendieck Conjecture” is known to hold in this situation [cf. [Mzk14], Corollary 2.3], we thus conclude that φ H does indeed arise from an isomorphism of ∼ ﬁelds k α → k β , as desired. This completes the proof of assertion (iii). Remark 1.11.1. At the time of writing Corollary 1.11, (iii), constitutes the only absolute isomorphism version of the Grothendieck Conjecture over MLF’s [to the knowledge of the author] that may be applied to arbitrary hyperbolic curves. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 31 Section 2: Geometric Uniformly Toral Neighborhoods In the present §2, we prove a certain “resolution of nonsingularities” type re- sult [cf. Lemma 2.6; Remark 2.6.1; Corollary 2.11] — i.e., a result reminiscent of the main results of [Tama2] [cf. also the techniques applied in the veriﬁcation of “observation (iv)” given in the proof of [Mzk9], Corollary 3.11] — that allows us to apply the theory of uniformly toral neighborhoods developed in [Mzk15], §3, to prove a certain “conditional absolute p-adic version of the Grothendieck Conjec- ture” — namely, that “point-theoreticity implies geometricity” [cf. Corollary 2.9]. This condition of point-theoreticity may be removed if, instead of starting with a hyperbolic curve, one starts with a “pro-curve” obtained by removing from a proper curve some [necessarily inﬁnite] set of closed points which is “p-adically dense in a Galois-compatible fashion” [cf. Corollary 2.10]. First, we recall the following “positive slope version of Hensel’s lemma” [cf. [Serre], Chapter II, §2.2, Theorem 1, for a discussion of a similar result]. Lemma 2.1. (Positive Slope Version of Hensel’s Lemma) Let k be a complete discretely valued ﬁeld; O k ⊆ k the ring of integers of k [equipped with the topology determined by the valuation]; m k the maximal ideal of O k ; π ∈ m k a def def uniformizer of O k ; A = O k [[X 1 , . . . , X m ]]; B = O k [[Y 1 , . . . , Y n ]]. Let us suppose that A (respectively, B) is equipped with the topology determined by its maximal def def ideal; write X = Spf(A) (respectively, Y = Spf(B)), K A (respectively, K B ) for the quotient ﬁeld of A (respectively, B), and Ω A (respectively, Ω B ) for the module of continuous diﬀerentials of A (respectively, B) over O k [so Ω A (respectively, Ω B ) is a free A- (respectively, B-) module of rank m (respectively, n)]. Let φ : B → A be the continuous O k -algebra homomorphism induced by an assignment B Y j → f j (X 1 , . . . , X m ) ∈ A [where j = 1, . . . , n]; let us suppose that the induced morphism dφ : Ω B ⊗ B A → Ω A satisﬁes the property that the image of dφ ⊗ A K A is a K A -subspace of rank n in Ω A ⊗ A K A [so n ≤ m]. Then there exists a point β 0 ∈ Y(O k ) and a positive integer r satisfying the following property: Let k be a ﬁnite extension of k, with ring of integers O k ; write B(β 0 , k , r) for the “ball” of points β ∈ Y(O k ) such that β , β 0 map to the same point of Y(O k /(π r )). Then the image of the map X (O k ) → Y(O k ) induced by φ contains the “ball” B(β 0 , k , r). Proof. First, let us observe that by Lemma 2.2 below, after possibly re-ordering the X i ’s, we may assume that the diﬀerentials dX i ∈ Ω A , where i = n + 1, . . . , m, together with the diﬀerentials df j ∈ Ω A , where j = 1, . . . , n, form a K A -basis of Ω A ⊗ A K A . Thus, by adding indeterminates Y n+1 , . . . , Y m to B and extending φ by sending Y i → X i for i = n + 1, . . . , m, we may assume without loss of generality that n = m, A = B, X = Y, i.e., that the morphism Spf(φ) : X → Y = X is “generically formally étale”. 32 SHINICHI MOCHIZUKI Write M for the n by n matrix with coeﬃcients ∈ A given by {df i /dX j } i,j=1,... ,n ; g ∈ A for the determinant of M . Thus, by elementary linear algebra, it follows that there exists an n by n matrix N with coeﬃcients ∈ A such that M ·N = N ·M = g·I [where we write I for the n by n identity matrix]. By our assumption concerning the image of dφ⊗ A K A , it follows that g = 0, hence, by Lemma 2.3 below, that there def exist elements x i ∈ m k , where i = 1, . . . , n, such that g 0 = g(x 1 , . . . , x n ) ∈ m k is nonzero. By applying appropriate “aﬃne translations” to the domain and codomain of φ, we may assume without loss of generality that, for i = 1, . . . , n, we have def def x i = f i (0, . . . , 0) = 0 ∈ O k . Write M 0 = M (0, . . . , 0), N 0 = N (0, . . . , 0) [so M 0 , N 0 are n by n matrices with coeﬃcients ∈ O k ]. Next, suppose that g 0 ∈ m sk \m s+1 k . In the remainder of the present proof, all “vectors” are to be understood as column vectors with coeﬃcients ∈ O k , where k is as in the statement of Lemma 2.1. Then I claim that for every vector y = (y 1 , . . . , y n ) ≡ 0 (mod π 3s ), there exists a vector x = (x 1 , . . . , x n ) ≡ 0 (mod π 2s ) def such that f (x ) = (f 1 (x ), . . . , f n (x )) = y . Indeed, since O k is complete and x i = f i (0, . . . , 0) = 0, it suﬃces [by taking x {2} to be (0, . . . , 0)] to show, for each integer l ≥ 2, that the existence of a vector x {l} ≡ 0 (mod π 2s ) such that f (x {l}) ≡ y (mod π (l+1)s ) implies the existence of a vector x {l + 1} such that x {l + 1} ≡ x {l} def (mod π ls ), f (x {l + 1}) ≡ y (mod π (l+2)s ). To this end, we compute: Set = def def def y − f (x {l}), η = g 0 −1 · , δ = N 0 ·η , x {l + 1} = x {l}+ δ. Thus, ≡ 0 (mod π (l+1)s ), η ≡ 0 (mod π ls ), δ ≡ 0 (mod π ls ), x {l + 1} ≡ x {l} (mod π ls ). In particular, the squares of the elements of δ all belong to π (l+2)s · O k [since l ≥ 2], so we obtain that f (x {l + 1}) ≡ f (x {l} + δ) ≡ f (x {l}) + M 0 · δ (mod π (l+2)s ) ≡ y − + M 0 · N 0 · η ≡ y − + g 0 · η ≡ y (mod π (l+2)s ) — as desired. This completes the proof of the claim. On the other hand, one veriﬁes immediately that the content of this claim is suﬃcient to complete the proof of Lemma 2.1. Remark 2.1.1. Thus, the usual “(slope zero version of ) Hensel’s lemma” corre- sponds, in the notation of Lemma 2.1, to the case where the image of the morphism dφ is a direct summand of Ω A . In this case, we may take r = 1. Remark 2.1.2. In fact, according to oral communication to the author by F. Oort, it appears that the sort of “positive slope version” of “Hensel’s lemma” given in Lemma 2.1 [i.e., where the derivative is only generically invertible] preceded the “slope zero version” that is typically referred to “Hensel’s lemma” in modern treatments of the subject. Lemma 2.2. (Subspaces and Bases of a Vector Space) Let k be a ﬁeld; V a ﬁnite-dimensional k-vector space with basis {e i } i∈I ; W ⊆ V a k-subspace. Then there exists a subset J ⊆ I such that if we write V J ⊆ V for the k-subspace generated by the e j , for j ∈ J, then the natural inclusions V J → V , W → V determine an ∼ isomorphism V J ⊕ W → V of k-vector spaces. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 33 Proof. This result is a matter of elementary linear algebra. Lemma 2.3. (Nonzero Values of Functions Deﬁned by Power Series) Let k, O k , A be as in Lemma 2.1; f = f (X 1 , . . . , X m ) ∈ A a nonzero element. Then there exist elements x i ∈ m k , where i = 1, . . . , m, such that f (x 1 , . . . , x m ) ∈ m k is nonzero. Proof. First, I claim that by induction on m, it suﬃces to verify Lemma 2.3 when m = 1. Indeed, for arbitrary m ≥ 2, one may write f = ∞ i c i X m i=0 — where c i = c i (X 1 , . . . , X m−1 ) ∈ O k [[X 1 , . . . , X m−1 ]]. Since f = 0, it follows that there exists at least one nonzero c j . Thus, by the induction hypothesis, it follows that there exist x i ∈ m k , where i = 1, . . . , m−1, such that m k c j (x 1 , . . . , x m−1 ) = 0. Thus, f (x 1 , . . . , x m−1 , X m ) ∈ O k [[X m ]] is nonzero, so, again by the induction hypothesis, there exists an x m ∈ m k such that m k f (x 1 , . . . , x m−1 , x m ) = 0. This completes the proof of the claim. def Thus, for the remainder of the proof, we assume that m = 1 and write X = X 1 , f = ∞ c i X i i=0 — where c i ∈ O k . Suppose that c j = 0, but that c i = 0 for i < j. Then there exists a positive integer s such that c j ∈ m sk . Let x ∈ m sk be any nonzero element. Then c j x j ∈ x j · m sk , while c i x i ∈ x j · x · O k ⊆ x j · m sk for any i > j. But this implies that m k f (x) = 0, as desired. In the following, we shall often work with [two-dimensional] log regular log schemes. For various basic facts on log regular log schemes, we refer to [Kato]; [Mzk2], §1. If X log is a log regular log scheme, then for integers j ≥ 0, we shall write [j] U X ⊆ X for the j-interior of X log , i.e., the open subscheme of points at which the ﬁber of the groupiﬁcation of the characteristic sheaf of X log is of rank ≤ j [cf. [MT], [j] Deﬁnition 5.1, (i); [MT], Proposition 5.2, (i)]. Thus, the complement of U X in X def [0] is a closed subset of codimension > j [cf. [MT], Proposition 5.2, (ii)]; U X = U X is the interior of X log [i.e., the maximal open subscheme over which the log structure is trivial]. Also, we shall write D X ⊆ X for the closed subscheme X\U X with the reduced induced structure. Finally, we remind the reader that in the following, all ﬁber products of fs log schemes are to be taken in the category of fs log schemes [cf. §0]. 34 SHINICHI MOCHIZUKI Next, let k be a complete discretely valued ﬁeld with perfect residue ﬁeld k; O k ⊆ k the ring of integers of k; k an algebraic closure of k; k the resulting algebraic closure of k; m k the maximal ideal of O k ; π ∈ m k a uniformizer of O k ; def X → S = Spec(O k ) def def a stable curve over S whose generic ﬁber X η = X × S η, where we write η = Spec(k), is smooth. Thus, X η is a proper hyperbolic curve over k, whose genus we denote by g X ; the open subschemes η ⊆ S, X η ⊆ X determine log regular log structures on S, X, respectively. We denote the resulting morphism of log schemes by X log → S log . Deﬁnition 2.4. (i) We shall refer to a morphism of log schemes φ log : V log → X log [or to the log scheme V log ] as a log-modiﬁcation if φ log admits a factorization V log → X log × S log S V log → X log def — where S V = Spec(O k V ), O k V is the ring of integers of a ﬁnite separable extension def k V of k; S V log is the log regular log scheme determined by the open immersion η V = Spec(k V ) → S V ; the morphism X log × S log S V log → X log is the projection morphism [where we observe that the underlying morphism of schemes X × S S V → S V is a stable curve over S V ]; the morphism V log → X log × S log S V log is a log étale morphism whose underlying morphism of schemes is proper and birational; we shall refer to k V as the base-ﬁeld of the log-modiﬁcation φ log . (ii) For i = 1, 2, let φ log : V i log → X log be a log-modiﬁcation that admits a i factorization V i log → X log × S log S i log → X log as in (i); ψ log : V 2 log → V 1 log an X log - morphism. Then let us observe that the log scheme V i log is always log regular of dimension 2 [cf. Proposition 2.5, (iv), below]. We shall refer to the log-modiﬁcation φ log as regular if the log structure of V i log is deﬁned by a divisor with normal i crossings [which implies that V i is a regular scheme]. We shall refer to the log- [1] modiﬁcation φ log as unramiﬁed if U V i is a smooth scheme over S i . We shall refer i to the morphism ψ log as a base-ﬁeld-isomorphism [or base-ﬁeld-isomorphic] if the morphism S 2 → S 1 induced by ψ log is an isomorphism. We shall refer to the points of V 1 over which the underlying morphism ψ of ψ log fails to be ﬁnite as the critical points of ψ log [or ψ]. We shall refer to the reduced divisor in V 2 determined by the closed set of points of V 2 at which ψ fails to be quasi-ﬁnite as the exceptional divisor of ψ log [or ψ]. We shall refer to the log scheme- (respectively, scheme-) theoretic ﬁber of V i log (respectively, V i ) over the unique closed point of S i as the log special ﬁber (respectively, special ﬁber) of V i log (respectively, V i ); we shall use the notation V log (respectively, V i ) i TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 35 to denote the log special ﬁber (respectively, special ﬁber) of V i log (respectively, V i ). Suppose that C is an irreducible component of V i . Then we shall say that C is stable if it maps ﬁnitely to X via φ i ; we shall say that the log-modiﬁcation φ log as i [1] unramiﬁed at C if U V i is smooth over S i at the generic point of C. Remark 2.4.1. Recall that there exists a base-ﬁeld isomorphic log-modiﬁcation V log → X log which is uniquely determined up to unique isomorphism [over X] by the follow- ing two properties: (a) V is regular; (b) V → S is a semi-stable curve. In fact, it was this example that served as the primary motivating example for the au- thor in developing the notion of a “log-modiﬁcation”. Note, moreover, that unlike property (a), however, the principal condition that deﬁnes a [base-ﬁeld-isomorphic] “log-modiﬁcation” — i.e., the condition that the morphism V → X be a proper, bi- rational morphism that extends to a log étale morphism V log → X log of log schemes — is a condition on the morphism V → X that has the virtue of being manifestly stable under base-change [i.e., via morphisms that satisfy suitable, relatively mild conditions]. This property of stability under base-change will be applied repeatedly in the remainder of the present §2. Proposition 2.5. (First Properties of Log-modiﬁcations) For i = 1, 2, let : V i log → X log φ log i be a log-modiﬁcation that admits a factorization V i log → X log × S log S i log → X log as in Deﬁnition 2.4, (i); ψ log : V 2 log → V 1 log an X log -morphism. Write S i = Spec(O k i ), where, for simplicity, we assume that the extension k i of k is a subﬁeld of k; k i for the residue ﬁeld of k i ; U ψ noncr ⊆ V 1 for the open subscheme given by the complement of the critical points of ψ log . Let • ∈ {1, 2}. Then: [1] (i) (The Noncriticality of the 1-Interior) We have: U V 1 ⊆ U ψ noncr . (ii) (Isomorphism over the Noncritical Locus) The morphism V 2 log → V 1 log × S log S 2 log determined by ψ log is an isomorphism over U ψ noncr . 1 (iii) (Log Smoothness and Unramiﬁed Log-modiﬁcations) V • log is log smooth over S • log . In particular, the sheaf of relative logarithmic diﬀerentials of the morphism V • log → S • log is a line bundle, which we shall denote ω V log /S log ; • • we have a natural isomorphism ψ ∗ ω V log /S log ∼ = ω V log /S log . Finally, there exists a 1 1 2 2 def ﬁnite extension k ◦ of k • such that, if we write S ◦ = Spec(O k ◦ ), then the mor- def phism V ◦ log = V • log × S log S ◦ log → X log determined by φ log • , is an unramiﬁed log- • modiﬁcation. [1] (iv) (Regularity and Log Regularity) V • log is log regular; U V • is regu- lar. Moreover, there exists a regular log-modiﬁcation V ◦ log → X log that admits 36 SHINICHI MOCHIZUKI a base-ﬁeld-isomorphic X log -morphism V ◦ log → V • log such that every irreducible component C of the special ﬁber V ◦ is smooth over the residue ﬁeld k ◦ of the base-ﬁeld k ◦ of V ◦ log . Finally, if the log-modiﬁcation V ◦ log → X log is unramiﬁed at such an irreducible component C, and we write D C ⊆ C for the reduced divisor [1] determined by the complement of C U V ◦ in C, then we have a natural isomor- phism ∼ ω C/k ◦ (D C ) → ω V log /S log | C ◦ ◦ of line bundles on C. is a regular log- (v) (Chains of Projective Lines) Suppose that φ log 2 modiﬁcation [so D V 2 is a divisor with normal crossings]. Then, after possibly replacing k 2 by a ﬁnite unramiﬁed extension of the discretely valued ﬁeld k 2 , ev- ery irreducible component C of D V 2 that lies in the exceptional divisor of ψ log is isomorphic to the projective line over the residue ﬁeld k(c) of the point c of V 1 log × S log S 2 log to which C maps. Moreover, C meets the other irreducible com- ponents of D V 2 at precisely two k(c)-valued points of C. That is to say, every connected component of the exceptional divisor of ψ log is a “chain of P 1 ’s”. (vi) (Dual Graphs of Special Fibers) The spectrum V x of the local ring obtained by completing the geometric special ﬁber V • × k • k at any point x which does not (respectively, does) belong to the 1-interior has precisely two (respec- tively, precisely one) irreducible component(s). In particular, the special ﬁber V • determines a dual graph Γ V • , whose vertices correspond bijectively to the irreducible components of V • × k • k, and whose edges correspond bijectively to the [1] points of (V • \U V • ) × k • k [so each edge abuts to the vertices corresponding to the irreducible components in which the point corresponding to the edge lies]. In dis- cussions of Γ V • , we shall frequently identify the vertices and edges of Γ V • with the corresponding irreducible components and points of V • × k • k. If the natural Galois action of Gal(k/k • ) on Γ V • is trivial, then we shall say that V • log is split. Finally, def the loop-rank lp-rk(V • ) = lp-rk(Γ V • ) [cf. §0] is equal to the loop-rank lp-rk(X). (vii) (Filtered Projective Systems) Given any log-modiﬁcation V ◦ log → log X log , there exists a log-modiﬁcation V •◦ → X log that admits X log -morphisms log log log log V •◦ → V • , V •◦ → V ◦ . That is to say, the log-modiﬁcations over X log form a ﬁltered projective system. (viii) (Functoriality) Let Y → S be a stable curve, with smooth generic def ﬁber Y η = Y × S η; Y log the log regular log scheme determined by the open sub- scheme Y η ⊆ Y . Then every ﬁnite morphism Y η → X η extends to a commutative diagram W • log −→ V • log ⏐ ⏐ ⏐ ⏐ Y log −→ X log — where W • log → Y log is a log-modiﬁcation. If lp-rk(Y ) = lp-rk(X), then we shall say that the morphisms Y η → X η , Y log → X log , W • log → V • log are loop- preserving; if lp-rk(Y ) > lp-rk(X) [or, equivalently, lp-rk(Y ) = lp-rk(X)], then TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 37 we shall say that the morphisms Y η → X η , Y log → X log , W • log → V • log are loop- ifying. Let C be an irreducible component of W • . Then we shall refer to C as base-stable [relative to Y η → X η ] (respectively, base-semi-stable [relative to W • log → V • log ]) if it maps ﬁnitely to a(n) stable (respectively, arbitrary) irreducible component of V • . If there exist log-modiﬁcations W ◦ log → Y log , V ◦ log → X log that ﬁt into a commutative diagram W ◦ log ⏐ ⏐ −→ V ◦ log ⏐ ⏐ W • log −→ V • log [where the left-hand vertical arrow is a Y log -morphism; the right-hand vertical arrow is an X log -morphism] such that C is the image of a base-semi-stable [relative to W ◦ log → V ◦ log ] irreducible component of W ◦ , then we shall say that C is potentially base-semi-stable [relative to Y η → X η ]. (ix) (Centers in the 1-Interior) Let k ◦ be a ﬁnite separable extension of k; K ◦ a discretely valued ﬁeld containing k ◦ which induces an inclusion O k ◦ ⊆ O K ◦ ∼ × between between the respective rings of integers and a bijection k ◦ /O k × ◦ → K ◦ /O K ◦ the respective value groups; x ◦ ∈ X(K ◦ ) a K ◦ -valued point. Then there exists an unramiﬁed log-modiﬁcation V ◦ log → X log with base-ﬁeld k ◦ such that the def morphism V ◦ → S ◦ = Spec(O k ◦ ) is a semi-stable curve, and x ◦ extends to a [1] point ∈ U V ◦ (O K ◦ ). (x) (Maps to the Jacobian) Suppose [for simplicity] that φ log is a base- • [1] ﬁeld-isomorphism. Let x • ∈ U V • (O k ); C the [unique, by (vi)] irreducible compo- [1] def nent of V • that meets [the image of ] x • ; F x • = V • \(C U V • ) ⊆ V • [regarded as a def closed subset]; U x • = V • \F x • ⊆ V • [so the image of x • lies in U x • ]. Write J η → η for the Jacobian of X η ; J → S for the uniquely determined semi-abelian scheme over S that extends J η ; ι η : X η → J η for the morphism that sends a T - valued point ξ [where T is a k-scheme] of X η , regarded as a divisor on X η × k T , to the point of J η determined by the degree zero divisor ξ − (x • | T ). Then ι η extends uniquely to a morphism U x • → J. If, moreover, X is loop-ample [cf. §0], then this morphism U x • → J is unramiﬁed. (xi) (Lifting Simple Paths) In the situation of (viii), suppose further that the following conditions hold: (a) the log-modiﬁcations W • log → Y log , V • log → X log are base-ﬁeld-isomorphic and split [cf. (vi)]; (b) the morphism W • log → V • log is ﬁnite. Let γ V be a simple path [cf. §0] in the dual graph Γ V • of V • [cf. (vi)]. Then there exists a simple path γ W in the dual graph Γ W • of W • that lifts γ V in the ∼ sense that the morphism W • → V • induces an isomorphism of graphs γ W → γ V . Suppose further that the following condition holds: (c) the morphism W • log → V • log is loop-preserving. 38 SHINICHI MOCHIZUKI Then γ W is unique in the sense that if γ W is any simple path in Γ W • that lifts γ V and is co-terminal [cf. §0] with γ W , then γ W = γ W . (xii) (Loop-preservation and Wild Ramiﬁcation) In the situation of (xi), suppose that, in addition to the conditions (a), (b), (c) of (xi), the following con- ditions hold: (d) there exists a prime number p such that k is of characteristic p, and the morphism Y η → X η is ﬁnite étale Galois and of degree p; (e) the morphism Y η → X η is wildly ramiﬁed over the terminal vertices [cf. §0] of the simple path γ V . Let w exc be a vertex of Γ W • that corresponds to an irreducible component of the exceptional divisor of W • log → Y log [i.e., a non-stable irreducible component of W • ] and, moreover, maps to a vertex v exc of Γ V • lying in γ V . Then the morphism Y η → X η is wildly ramiﬁed at w exc . Proof. First, we consider assertion (i). We may assume without loss of generality that ψ is a base-ﬁeld-isomorphism. Then it follows from the simple structure of the [1] monoid N that any log étale birational morphism over U V 1 is, in fact, étale. This completes the proof of assertion (i). Next, we consider assertion (iii). Since the morphism X log → S log is log smooth, and the morphism V • log → X log × S log S • log is log étale, we conclude that V • log is log smooth over S • log ; the portion of assertion (iii) concerning ω V log /S log then follows immediately. To verify the portion of assertion • • (iii) concerning unramiﬁed log-modiﬁcations, it suﬃces to observe that, in light of the well-known local structure of the nodes of the stable log curve X log × S log S • log → def S • log , there exists a ﬁnite extension k ◦ of k • such that, if we write S ◦ = Spec(O k ◦ ), def then the morphism V ◦ log = V • log × S log S ◦ log → S ◦ log admits sections that intersect • [1] with every irreducible component of V ◦ U V ◦ ; thus, the fact that V ◦ log → X log is an unramiﬁed log-modiﬁcation follows immediately from the log smoothness of [1] U V • , in light of the simple structure of the monoid N. This completes the proof of assertion (iii). Next, we consider assertion (iv). The fact that V • log is log regular follows immediately from the log smoothness of V • log over S • log [cf. assertion (iii)]; the fact [1] [1] that U V • is regular then follows from the log regularity of U V • , in light of the simple structure of the monoid N. To construct a regular log-modiﬁcation V ◦ log → X log that admits a base-ﬁeld-isomorphic X log -morphism V ◦ log → V • log , it suﬃces to “resolve [1] the singularities” at the ﬁnitely many points of V • \U V • . To give a “resolution of singularities” of the sort desired, it suﬃces to construct, for each such v, a “fan” arising from a “locally ﬁnite nonsingular subdivision of the strongly convex rational polyhedral cone associated to the stalk of the characteristic sheaf of V • log at v that is equivariant with respect to the Galois action on the stalk” [cf., e.g., the discussion at the beginning of [Mzk2], §2]. Since this is always possible [cf., e.g., the references quoted in the discussion of loc. cit.], we thus obtain a regular log-modiﬁcation V ◦ log → X log that admits a base-ﬁeld-isomorphic X log -morphism V ◦ log → V • log ; moreover, by replacing V ◦ log with the result of blowing up once more at [1] various points of V ◦ \U V ◦ , we may assume that each irreducible component C of V ◦ TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 39 is smooth over k ◦ , as desired. Finally, the construction of the natural isomorphism [1] ∼ ω C/k ◦ (D C ) → ω V log /S log | C is immediate over C U V ◦ [cf. our assumption that the ◦ ◦ log-modiﬁcation V ◦ log → X log is unramiﬁed at C!]; one may then extend this natural isomorphism to C by means of an easy local calculation at the points of D C . This completes the proof of assertion (iv). By assertions (iii) and (iv), the underlying schemes of the domain and codomain of the morphism V 2 log → V 1 log × S log S 2 log of 1 assertion (ii) are normal. Thus, assertion (ii) follows immediately from Zariski’s main theorem. Next, we consider assertion (v). We may assume without loss of generality are base-ﬁeld-isomorphic [so ψ log is log étale]. Also, by blowing up that the φ log i [1] once more at various points of V 2 \U V 2 [cf. the proof of assertion (iv)], one veriﬁes immediately that we may assume without loss of generality that C is smooth over k 2 . Let us write C log for the log scheme obtained by equipping C with the log structure determined by the points of C that meet the other irreducible components of D V 2 . Thus, after possibly replacing k 2 by a suitable ﬁnite unramiﬁed extension of k 2 , we may assume that the interior U C ⊆ C of C log is the open subscheme of a smooth proper curve of genus g over k(c) obtained by removing a divisor D C ⊆ C of degree r > 0 over k(c). On the other hand, it follows immediately from the deﬁnition of a log-modiﬁcation [cf. Deﬁnition 2.4, (i)] and the well-known general theory of toric varieties [cf. the discussion of “fans” in the proof of assertion (iv)] that the pair (C, D C ) determines a toric variety of dimension one, and hence that U C is isomorphic to a copy of G m over k(c), i.e., that g = 0, r = 2, as desired. Now the remaining portions of assertion (v) follow immediately. This completes the proof of assertion (v). Next, we consider assertion (vi). First, we observe that if x belongs to the 1-interior, then it follows immediately from the simple structure of the monoid N [and the deﬁnition of the 1-interior] that V x is irreducible. Thus, it suﬃces to consider the case where x does not belong to the 1-interior. Let V ◦ log → X log , V ◦ log → V • log be as in assertion (iv). We may assume without loss of generality that the log-modiﬁcations V • log → X log , V ◦ log → X log are base-ﬁeld-isomorphisms. Also, by replacing k be a ﬁnite unramiﬁed extension of k, we may assume that every irreducible component of V • is geometrically irreducible over k, and that every point [1] of V • \U V • is deﬁned over k. Then, since V ◦ log → V • log is birational, and V • log is log regular, hence, in particular, normal [cf. assertion (iv)], it follows from Zariski’s [1] main theorem that the points of V • \U V • correspond precisely [via V ◦ log → V • log ] to [1] the connected components of the inverse image of V • \U V • via V ◦ log → V • log . Thus, the remainder of assertion (vi) follows immediately from assertion (v), applied to V ◦ log → V • log , V ◦ log → X log . This completes the proof of assertion (vi). Next, we consider assertion (vii). We may assume without loss of generality that the log-modiﬁcations V • log → X log , V ◦ log → X log are base-ﬁeld-isomorphic. log def Then to verify assertion (vii), it suﬃces to observe that one may take V •◦ = log log V • × X log V ◦ . This completes the proof of assertion (vii). In a similar vein, assertion (viii) follows by observing that the fact that Y η → X η extends to a morphism Y log → X log follows, for instance, from [Mzk2], Theorem A, (1); thus, def one may take W • log = V • log × X log Y log . This completes the proof of assertion (viii). 40 SHINICHI MOCHIZUKI Next, we consider assertion (ix). We may assume without loss of generality that k = k ◦ . Let V ◦ log → X log be the base-ﬁeld-isomorphic unramiﬁed log-modiﬁcation determined by the regular semi-stable model of X over S [cf. Remark 2.4.1]. Since V ◦ is proper over S, it follows that x ◦ extends to a point ∈ V ◦ (O K ◦ ); if this point [1] fails to lie in U V ◦ , then it follows that it meets one of the nodes of V ◦ . On the other hand, since [after possibly replacing k ◦ by a ﬁnite unramiﬁed extension of the discretely valued ﬁeld k ◦ ] the completion of the regular scheme V ◦ at such a node is necessarily isomorphic over O k ◦ to a complete local ring of the form O k ◦ [[s, t]]/(s · t − π ◦ ) [where s, t are indeterminates; π ◦ is a uniformizer of O k ◦ ], this contradicts our ∼ × assumption that O k ◦ ⊆ O K ◦ induces a bijection k ◦ /O k × ◦ → K ◦ /O K [i.e., by con- ◦ sidering the images via pull-back by x ◦ of s, t in these value groups, in light of the relation “s · t − π ◦ ”]. This completes the proof of assertion (ix). Next, we consider assertion (x). Write N → S for the Néron model of J η over S. Thus, J may be regarded as an open subscheme of N . Note that the existence of the rational point x • implies that U x • is smooth over S [cf. the proof of assertion (iii)]. Thus, it follows from the universal property of the Néron model [which is typically used to deﬁne the Néron model] that ι η extends to a morphism U x • → N . Since C U x • is connected [cf. the deﬁnition of U x • ], the fact that the image of this morphism lies in J ⊆ N follows immediately from the fact that [by deﬁnition] it maps x • to the identity element of J(O k ). Thus, we obtain a morphism U x • → J. To verify that this morphism is unramiﬁed, it suﬃces [by considering appropriate translation automorphisms of J] to show that it induces a surjection on Zariski cotangent spaces at x • ; but the induced map on Zariski cotangent spaces at x • is easily computed [by considering the long exact sequence on cohomology associated to the short exact sequence 0 → O V • → O V • (x • ) → O V • (x • )| x • → 0 on V • , then taking duals] to be the map ∼ H 0 (X log , ω X log /S log ) → H 0 (V log , ω V log /S log ) → ω V log /S log | x • • • ∼ [where we recall the natural isomorphism ω X log /S log | V log → ω V log /S log , arising from • • log → X log is log étale — cf. assertion (iii)] given by evaluating the fact that φ log • : V • at x • , hence is surjective so long as X is loop-ample [cf. §0]. This completes the proof of assertion (x). Next, we consider assertion (xi). First, let us observe that it follows immedi- ately from the surjectivity of W • → V • that every vertex of Γ V • may be lifted to a vertex of Γ W • . Next, I claim that every edge of Γ V • may be lifted to an edge of Γ W • . Indeed, let y ∈ W • (k), x ∈ V • (k) be such that y → x; write W y , V x for the respective spectra of the local rings obtained by completing W • , V • at y, x. Then the morphism W log → V log induces a ﬁnite, dominant, hence surjective, morphism W y → V x . In particular, this morphism W y → V x induces a surjection from the set I y of irreducible components of W • that pass through y to the set I x of irreducible components of V • that pass through x. Thus, if x corresponds to an edge of the dual graph Γ V • , then this set I x is of cardinality 2 [cf. assertion (vi)]; since I y is of cardinality ≤ 2 [cf. assertion (vi)], the existence of the surjection I y I x thus TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 41 ∼ implies that this surjection is, in fact, a bijection I y → I x , hence that y corresponds to an edge of Γ W • [cf. assertion (vi)]. This completes the proof of the claim. Thus, by starting at one of the terminal vertices of γ V , and proceeding along γ V from vertex to edge to vertex, etc., one concludes immediately the existence of a simple path γ W lifting γ V . To verify uniqueness when condition (c) holds, write v 1 , v 2 for the terminal vertices of γ V ; e for the edge of γ V that is nearest to v 1 among those ; v 1 (respectively, v 2 ) for the vertex edges of γ V that lift to diﬀerent edges in γ W , γ W to which e abuts that lies in the same connected component of the complement of e in γ V as v 1 (respectively, v 2 ); v 1 + for the vertex of γ V that is nearest to v 1 among those vertices of γ V lying between v 2 and v 2 which lift to the same vertex in γ W , . Then by traveling along γ W from the vertex w 1 of γ W lifting v 1 to the vertex γ W + w 1 of γ W lifting v 1 + , then traveling back along γ W from w 1 + [which, by deﬁnition, ] to w 1 [which, by deﬁnition, also belongs to γ W ], one obtains also belongs to γ W a “nontrivial loop” in Γ W • [i.e., a nonzero element of H 1 (Γ W • , Q)] that maps to a “trivial loop” in Γ V • [i.e., the zero element of H 1 (Γ V • , Q)]. But this contradicts the assumption that the morphism W • log → V • log is loop-preserving [cf. condition (c)]. This completes the proof of assertion (xi). Finally, we consider assertion (xii). First, we observe that the hypotheses of assertion (xii) are stable with respect to base-change in S. In particular, we may always replace S = Spec(O k ) by the normalization of S in some ﬁnite separable extension of k. Next, by assertion (iv), we may assume that there exists a base- ﬁeld-isomorphic, regular, split log-modiﬁcation V ◦ log → X log , together with an X log - def morphism V ◦ log → V • log . Moreover, if we take W ◦ log = V ◦ log × V log W • log , then • the composite morphism W ◦ log → W • log → Y log forms a base-ﬁeld-isomorphic log- modiﬁcation such that the projection W ◦ log → V ◦ log is ﬁnite [by condition (b); cf. also the ﬁniteness mentioned in the discussion entitled “Log Schemes” given in §0]. In particular, by replacing W • log → V • log by W ◦ log → V ◦ log [cf. also assertion (v), concerning the eﬀect on the simple path γ V ; assertion (vi), concerning the eﬀect on the loop-rank], we may assume that the log-modiﬁcation V • log → X log is regular. Here, let us note that since W • log → V • log is ﬁnite, and W • log is log regular [cf. assertion (iv)], which implies, in particular, that W • is normal [so W • is the def log normalization of V • in Y η ], it follows that G = Gal(Y η /X η ) ( ∼ = Z/pZ) acts on W • . Also, by assertion (iv), we may assume that there exists a base-ﬁeld-isomorphic, log log regular, split log-modiﬁcation W → Y log , together with a Y log -morphism W → W • log ; moreover, it follows immediately from the proof of assertion (iv) that we log log may choose W so that the action of G extends to W . Finally, we observe that it follows from assertion (v) that every irreducible component of the exceptional log divisors of W , V • [relative to the morphisms W → Y log , V • log → X log ] is isomorphic to P 1 k . Let E W be the irreducible component of W • corresponding to w exc . Thus, there exists a unique irreducible component F W of W that maps ﬁnitely to E W ; moreover, E W maps ﬁnitely to an irreducible component E V of V • [corresponding to v exc ] that lies in the exceptional divisor of V • log → X log . Thus, we have ﬁnite morphisms F W → E W → E V 42 SHINICHI MOCHIZUKI — where F W , E V are isomorphic to P 1 k ; the ﬁrst morphism F W → E W is a mor- phism between irreducible schemes that induces an isomorphism between the re- spective function ﬁelds. Now to complete the proof of assertion (xii), it suﬃces to assume that the composite morphism F W → E V is generically étale and derive a contradiction. Let us refer to the two k-valued points of F W (respec- [1] [1] tively, E V ) [cf. assertion (v)] that lie outside U W (respectively, U V • ) as the critical points of F W (respectively, E V ). Then since the divisor on W • log (respectively, V • log ) at which the morphism W • log → V • log is [necessarily wildly] ramiﬁed does not, by our assumption, contain E W (respectively, E V ), it follows that the divisor on F W (re- spectively, E V ) at which F W → E V is ramiﬁed is supported in the divisor deﬁned by the two critical points of F W (respectively, E V ). Next, let us write v 1 , v 2 for the terminal vertices of γ V . Note that by conditions (d), (e), it follows that v 1 , v 2 lift, respectively, to unique vertices w 1 , w 2 of Γ W • . In particular, it follows that any two simple paths in Γ W • lifting γ V are co-terminal. Now I claim that the morphism F W → E V is of degree p. Indeed, if this morphism of w exc such that is of degree 1, then it follows that there exists a G-conjugate w exc w exc = w exc . Thus, considering the G-conjugates of any simple path in Γ W • lifting γ V , it follows that we obtain two distinct [necessarily co-terminal!] simple paths in Γ W • lifting γ V . But this contradicts the uniqueness portion of assertion (xi). This completes the proof of the claim. Note that this claim implies that we have G-equivariant morphisms F W → E W → E V , where G acts trivially on E V . Next, I claim that G ﬁxes each of the critical points of F W . Indeed, it follows immediately from the deﬁnitions that G preserves the divisor of critical points of F W . Thus, if G fails to ﬁx each of the critical points of F W , then it follows that G permutes the two critical points of F W , hence that p = 2. But since F W → E V is of degree p and unramiﬁed outside the divisor of critical points of F W , this implies that P 1 k ∼ = F W → E V ∼ = P 1 k is ﬁnite étale, hence [since, as is well-known, the étale fundamental group of P 1 k is trivial!] that F W → E V is an isomorphism — in contradiction to the fact that F W → E V is of degree p > 1. This completes the proof of the claim. Note that this claim implies that the morphism F W → E V is ramiﬁed at the critical points of F W , and that the set of two critical points of F W maps bijectively to the set of two critical points of E V . In particular, it follows that F W → E V determines a ﬁnite étale covering (G m ) k → (G m ) k of degree p. On the other hand, any morphism (G m ) k → (G m ) k is determined by a unit on (G m ) k , i.e., by a k × -multiple of U n , where U is the standard coordinate on (G m ) k , and n is the degree of the morphism. Since the morphism determined by a k × -multiple of U p clearly fails to be generically étale, we thus obtain a contradiction. This completes the proof of assertion (xii). In the following, we shall write “π 1 (−)” for the “log fundamental group” of the log scheme in parentheses, relative to an appropriate choice of basepoint [cf. [Ill] for a survey of the theory of log fundamental groups]. Also, from now on, we shall assume, until further notice, that: TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 43 The discrete valuation ring O k is of mixed characteristic, with residue ﬁeld k perfect of characteristic p and of countable cardinality. Recall that by “Krasner’s lemma” [cf. [Kobl], pp. 69-70, as well as the proof given above of Lemma 2.1], given a splitting ﬁeld k over k of a monic polynomial f (T ) [where T is an indeterminate] of degree n with coeﬃcients in k, every monic polynomial h(T ) of degree n with coeﬃcients in k that are suﬃciently close [in the topology of k] to the coeﬃcients of f (T ) also splits in k . Thus, it follows from our assumption that k is of countable cardinality that k admits a countable collection F of subﬁelds which are ﬁnite and Galois over k such that every ﬁnite Galois extension of k contained in k is contained in a subﬁeld that belongs to the collection F. In some sense, the main technical result of the present §2 is the following lemma. Lemma 2.6. (Prime-power Cyclic Coverings and Log-modiﬁcations) Suppose that X is loop-ample [cf. §0]. Then: (i) (Existence of Wild Ramiﬁcation) Let X η + → X η be a ﬁnite étale Galois covering of hyperbolic curves over η with stable reduction over S such that Gal(X η + /X η ) is isomorphic to a product of 2g X copies of Z/pZ [so such a covering always exists after possibly replacing k by a ﬁnite extension of k]; V log → X log a split, base-ﬁeld-isomorphic log-modiﬁcation. Then X η + → X η is wildly ramiﬁed over every irreducible component C of V . (ii) (Loopiﬁcation vs. Component Crushing) After possibly replacing k by a ﬁnite extension of k, there exist data as follows: a stable curve Y → S with def smooth generic ﬁber Y η = Y × S η and associated log scheme Y log ; a cyclic ﬁnite étale covering Y η → X η of degree a positive power of p — which determines a morphism Y log → X log — such that at least one of the following two conditions is satisﬁed: (a) Y η → X η is loopifying and wildly ramiﬁed at some [necessarily] stable irreducible component C of Y which is potentially base-semi- stable relative to Y η → X η ; (b) there exists a [necessarily] stable irreducible component C of Y which is not potentially base-semi-stable relative to Y η → X η . (iii) (Components Crushed to the 1-Interior) In the situation of (ii), there exists a commutative diagram W log ⏐ ⏐ −→ Q log ⏐ ⏐ Y log −→ X log — where the vertical morphisms are split, base-ﬁeld-isomorphic log-modiﬁcations; the horizontal morphism in the bottom line is the morphism already referred to; the 44 SHINICHI MOCHIZUKI natural action of Gal(Y η /X η ) on Y log extends to W log — such that the following property is satisﬁed: If condition (a) (respectively, (b)) of (ii) is satisﬁed, then the unique irreducible component C W of W that maps ﬁnitely to the irreducible compo- nent C of condition (a) (respectively, (b)) maps ﬁnitely to Q (respectively, maps [1] to a closed point of U Q ). (iv) (Group-theoretic Characterization of Crushing) In the situation of (ii), let C be a [necessarily] stable irreducible component of Y ; l a prime = p. Write Gal(k/k) G k log for the maximal tamely ramiﬁed quotient; Δ Y log (respectively, Δ X log ) for the maximal pro-l quotient of the kernel of the natural [outer] surjection π 1 (Y log ) G k log (respectively, π 1 (X log ) G k log ); Δ C ⊆ Δ Y log for the decomposition group of C [well-deﬁned up to conjuga- tion by an element of π 1 (Y log )]. [Thus, Δ Y log (respectively, Δ X log ) may be identi- ﬁed with the maximal pro-l quotient of Ker(π 1 (Y η ) Gal(k/k)) (respectively, Ker(π 1 (X η ) Gal(k/k))) — cf., e.g., [MT], Proposition 2.2, (v).] Then the fol- lowing two conditions are equivalent: (a) the image of Δ C in Δ X log is trivial; (b) there exists a commutative diagram W log ⏐ ⏐ −→ Q log ⏐ ⏐ Y log −→ X log — where the vertical morphisms are split, base-ﬁeld-isomorphic log-modi- ﬁcations; the horizontal morphism in the bottom line is the morphism already referred to — such that the unique irreducible component C W of [1] W that maps ﬁnitely to C maps to a closed point of U Q . (v) (Group-theoretic Characterization of Wild Ramiﬁcation) In the situation of (iv), the morphism Y η → X η is wildly ramiﬁed at C if and only if Gal(Y η /X η ) stabilizes [the conjugacy class of ] and induces the identity [outer automorphism] on the [normally terminal — cf. [Mzk13], Proposition 1.2, (ii); [Mzk17], Lemma 2.12] subgroup Δ C of Δ Y log . Proof. Let us write def T X = π 1 (X η × k k) ab ⊗ Z p for the maximal pro-p abelian quotient of the geometric fundamental group of X η . Thus, T X is a free Z p -module of rank 2g X . Next, we consider assertion (i). Upon base-change to k, the covering X η + → X η corresponds to the open subgroup p · T X ⊆ T X . Let us write J → S for the uniquely determined semi-abelian scheme that extends the Jacobian J η → η of X η . After possibly replacing k by a ﬁnite extension of k, there exists a rational [1] point x ∈ U V (O k ) that meets C. Thus, for some Zariski open neighborhood U x of the image of x in V , we obtain a morphism ι : U x → J, as in Proposition 2.5, (x). Moreover, since we have assumed that X is loop-ample, it follows that this morphism ι is unramiﬁed. Now if the morphism X η + → X η is tamely ramiﬁed over TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 45 C, then it follows from our assumptions on the covering X η + → X η , together with the interpretation of J in terms of Néron models [cf. the proof of Proposition 2.5, (x)], that [after possibly replacing k by a ﬁnite extension of k] there exists some ﬁnite separable extension L of the function ﬁeld k(C) of C such that there exists a commutative diagram ι L −→ J Spec(L) ⏐ ⏐ ⏐ ⏐ [p] Spec(k(C)) ι C −→ J — where is the [étale] morphism determined by the given inclusion k(C) → L; [p] is the morphism given by multiplication by p on the group scheme J; ι C is the restriction of ι to Spec(k(C)). On the other hand, since the restriction of [p] to the special ﬁber J of J factors through the Frobenius morphism on J, it follows that [p] ◦ ι L fails to be unramiﬁed. Thus, since ι C ◦ is unramiﬁed, we obtain a contradiction to the commutativity of the diagram. This completes the proof of assertion (i). Next, we consider assertion (ii). If V log → X log is a split, base-ﬁeld-isomorphic log-modiﬁcation, and C is an irreducible component of V , then let us write D C ⊆ T X for the decomposition group associated to C and I C ⊆ D C for the wild inertia group associated to C. Note that since T X is abelian, and V log is split, it follows that the subgroups D C , I C are well-deﬁned [and completely determined by C]. Moreover, by assertion (i), it follows that I C has nontrivial image in T X ⊗ Z/pZ. Next, I claim that if V ◦ log → X log is a split, base-ﬁeld-isomorphic log-modiﬁcation, then V ◦ log → X log is completely determined, as a log scheme over X log , up to count- ably many possibilities, by X log . Indeed, the morphism V ◦ log → X log is an iso- morphism over the 1-interior of X log [cf. Proposition 2.5, (i), (ii)]. Moreover, at each of the ﬁnitely many points x of X log lying in the complement of the 1-interior, V ◦ log → X log is determined by countably many choices of certain combinatorial data involving the groupiﬁcation of the stalk of the characteristic sheaf of X log at x [cf. the proof of Proposition 2.5, (iv)]. This completes the proof of the claim. In partic- ular, since k is assumed to be of countable cardinality [cf. the discussion preceding the present Lemma 2.6], it follows that: There exists a countable coﬁnal collection M of split log-modiﬁcations of X log . In particular, it follows that if we write C for the set of all irreducible components of the special ﬁbers of log-modiﬁcations belonging to M, then the collection of [non- trivial] subgroups of T X of the form “I C ”, where C ∈ C, is of countable cardinality. Thus, we may, for instance, enumerate the elements of C via the natural numbers 46 SHINICHI MOCHIZUKI so as to obtain a sequence C 1 , C 2 , . . . [i.e., which includes all elements of C]. Since Z p , on the other hand, is of uncountable cardinality, we thus conclude that there exists a surjection Λ : T X Λ X — where Λ X ∼ = Z p — such that the following properties are satisﬁed: def (1) For every subgroup I C , where C ∈ C, we have Λ C = Λ(I C ) = {0}. (2) There exists a stable component C 0 of X such that Λ C 0 = Λ X . [For instance, by applying the fact that each I C n has nontrivial image in T X ⊗ Z/pZ, one may construct Λ by constructing inductively on n [a natural number] an increasing sequence of natural numbers m 1 < m 2 < . . . such that I C n maps to a nonzero subgroup of Λ X ⊗ Z/p m n Z.] Let us refer to a connected ﬁnite étale Galois covering X η → X η [which may only be deﬁned after possibly replacing k by a ﬁnite extension of k] of hyperbolic curves over η with stable reduction over S as a Λ-covering if the covering X η × k k → X η × k k arises from an open subgroup of Λ X ; thus, Gal(X η /X η ) may be thought of as a ﬁnite quotient of Λ X by an open subgroup Λ X ⊆ Λ X . Note that since every Λ C , where C ∈ C, is isomorphic to Z p , it follows that the following property also holds: (3) For any pair X η → X η → X η of Λ-coverings of X η such that Λ C ∼ has nontrivial image in Gal(X η /X η ) = Λ X /Λ X , it follows that Λ C Λ X surjects onto Λ X /Λ X — i.e., that the covering X η → X η is totally wildly ramiﬁed over any valuation of the function ﬁeld of X η whose center on X is equal to the generic point of C. Now to complete the proof of assertion (ii), it suﬃces to derive a contradiction upon making the following two further assumptions: (4) Every Λ-covering is loop-preserving. (5) For every Λ-covering X η → X η [which extends to a morphism (X ) log → X log of log stable curves], there exists [after possibly replacing k by a ﬁnite extension of k] a split, base-ﬁeld-isomorphic log-modiﬁcation V log → X log such that the morphism X η → X η ∼ = V η ⊆ V extends to a quasi-ﬁnite morphism from some Zariski neighborhood in X of the generic points of X to V . [Indeed, if assumption (4) is false, then it follows immediately that condition (a) of assertion (ii) holds [cf. property (2)]; if assumption (5) is false, then it follows imme- diately that condition (b) of assertion (ii) holds.] Note, moreover, that assumption (5) implies the following property: (6) For every Λ-covering X η → X η [which extends to a morphism (X ) log → X log of log stable curves], there exist [after possibly replacing k by a ﬁnite extension of k], split, base-ﬁeld-isomorphic log-modiﬁcations (V ) log → TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 47 (X ) log , V log → X log , together with a ﬁnite morphism (V ) log → V log lying over (X ) log → X log . def [Indeed, if, in the notation of property (5), one takes (V ) log = V log × X log (X ) log , then the natural projection morphism (V ) log → (X ) log is a split, base-ﬁeld- isomorphic log-modiﬁcation. Moreover, every irreducible component of V maps ﬁnitely either to some irreducible component of V or to some irreducible compo- nent of X . Thus, by property (5), we conclude that every irreducible component of V maps ﬁnitely to some irreducible component of V , hence [by Zariski’s main theorem] that the natural projection morphism (V ) log → V log is ﬁnite, as desired.] Next, let us observe that properties (4), (6) imply the following properties: (7) There exists a Λ-covering X η → X η [after possibly replacing k by a ﬁnite extension of k] such that every Λ C , where C ∈ C, has nontrivial image in Λ X /Λ X . (8) There exist Λ-coverings X η → X η → X η [after possibly replacing k by a ﬁnite extension of k] such that Λ X /Λ X ∼ = Z/pZ, and, moreover, for every Λ C , where C ∈ C, the intersection Λ C Λ X surjects onto Λ X /Λ X . Indeed, by property (3), it follows that property (8) follows immediately from prop- erty (7). To verify property (7), we reason as follows: First, let X η † → X η ∗ → X η be Λ-coverings [which exist after possibly replacing k by a ﬁnite extension of k] such that Λ X ∗ /Λ X † is of order p, and Λ C has nontrivial image in Λ X /Λ X ∗ [which implies that Λ C Λ X ∗ surjects onto Λ X ∗ /Λ X † — cf. property (3)] for every stable irreducible component C of X; write X †,log → X ∗,log → X log for the resulting morphisms of log stable curves. [Note that such Λ-coverings exist, precisely because there are only ﬁnitely many such stable C.] Let V †,log → X †,log , V ∗,log → X ∗,log , V log → X log be split, base-ﬁeld-isomorphic log-modiﬁcations such that there exist ﬁnite, loop-preserving morphisms V †,log → V ∗,log → V log lying over X †,log → X ∗,log → X log [cf. properties (4), (6)]. [Thus, V †,log , V ∗,log are completely determined by V log — i.e., by taking the normalization of V in X η † , X η ∗ .] Next, let us observe that every node ν of X determines a simple path γ V ν in the dual graph Γ V [i.e., by taking the inverse image of ν in V — cf. Proposition 2.5, (v)], whose terminal vertices are stable irreducible components of V [but whose non- terminal vertices are non-stable irreducible components of V ]. Thus, by Proposition 2.5, (xi) [which is applicable, in light of properties (4), (6)], it follows that γ V ν lifts [uniquely — i.e., once one ﬁxes liftings of the terminal vertices] to simple paths γ V ν ∗ in Γ V ∗ , γ V ν † in Γ V † . Since, moreover, X η † → X η ∗ is totally wildly ramiﬁed over the terminal vertices of γ V ν ∗ , it thus follows that we may apply Proposition 2.5, (xii), to conclude that X η † → X η ∗ is wildly ramiﬁed at every non-stable vertex of γ V ν † . Write B for the set of irreducible components of V [which we think of as valuations of the function ﬁeld of X η × k k] that are the images of stable vertices of γ V ν † , for nodes ν 48 SHINICHI MOCHIZUKI of X. Observe that if we keep the coverings X η † → X η ∗ → X η ﬁxed, but vary the log-modiﬁcation V log → X log [among, say, elements of M], then the set B remains unchanged [if we think of B as a set of valuations of the function ﬁeld of X η × k k] and of ﬁnite cardinality [bounded by the cardinality of the set of stable irreducible components of X † ]. Thus, in summary, if we think of B as a subset of C, then we may conclude the following: Λ C Λ X ∗ surjects onto Λ X ∗ /Λ X † , for all C ∈ C\B. Since B is ﬁnite, it thus follows that there exists a Λ-covering X η → X η † → X η such that Λ C Λ X † has nontrivial image in Λ X † /Λ X , for all C ∈ C [cf. property (3)]. This completes the proof of property (7). Next, let us consider Λ-coverings X η → X η → X [which exist after possibly replacing k by a ﬁnite extension of k] such that Λ X /Λ X is of order p, and Λ C has nontrivial image in Λ X /Λ X [which implies that Λ C Λ X surjects onto Λ X /Λ X — cf. property (3)] for all C ∈ C [cf. property (7)]; write (X ) log → (X ) log → X log for the resulting morphisms of log stable curves. Let (V ) log → (X ) log , (V ) log → (X ) log be split, base-ﬁeld-isomorphic, unramiﬁed log-modiﬁcations such that there exists a ﬁnite, loop-preserving morphism (V ) log → (V ) log lying over (X ) log → (X ) log [which exist after possibly replacing k by a ﬁnite extension of k — cf. properties (4), (6); Proposition 2.5, (iii)]. Next, let us consider the logarithmic derivative δ : ω (V ) log /S log | V → ω (V ) log /S log of the morphism (V ) log → (V ) log . Since this morphism is ﬁnite étale over η, it follows that δ is an isomorphism over η. On the other hand, since X η → X η is totally wildly ramiﬁed over every irreducible component C of V [i.e., induces a purely inseparable extension of degree p of the function ﬁeld of C], it follows that δ vanishes on the special ﬁber V . Write δ ∗ = π −n · δ, for the maximal integer n such that π −n · δ remains integral. Thus, we obtain a morphism def δ ∗ : ω (V ) log /S log | V → ω (V ) log /S log of line bundles on V which is not identically zero on V . Let C be an irreducible component of V such that δ ∗ | C ≡ 0. Note that since V has at least one stable irreducible component, it follows that we may choose C such that either C is stable or C meets an irreducible component C ∗ of V such that δ ∗ | C ∗ ≡ 0. Thus, if C is not stable, then it follows that δ ∗ | C has at least one zero [i.e., is not an isomorphism of line bundles]. Write C for the irreducible component of V which is the image of C ; E → C , E → C for the respective normalizations; g E , g E for the respective genera of E , E . Also, let us refer to the points of C , E , C , TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 49 E that do not map to the respective 1-interiors of (V ) log , (V ) log as critical points. Write D E ⊆ E , D E ⊆ E for the respective divisors of critical points; r E , r E for the respective degrees of D E , D E . Next, let us consider the morphism C → C . Since this morphism C → C induces a purely inseparable extension of degree p on the respective function ﬁelds, ∼ it follows that we have an isomorphism of k-schemes E × k k → E [so g E = g E ], where we write k → k ∼ = k for the [degree one!] ﬁeld extension determined by the Frobenius morphism on k. Next, I claim that the critical points of C map to critical points of C . Indeed, if a critical point of C maps to a non-critical point c of C , then let us write C c , C c for the spectra of the respective completions of the local rings of C , C at [the ﬁber over] c. Then observe that since V is regular [of dimension two] at c [cf. Proposition 2.5, (iv)], while V is the normalization of V in X η , it follows from elementary commutative algebra that V is ﬁnite and ﬂat over V of degree p at c. Thus, if we write η c for the spectrum of the residue ﬁeld of the unique generic point of the irreducible scheme C c , then C c × C c η c → η c is ﬁnite, ﬂat of degree ≤ p; [since X η → X η is totally wildly ramiﬁed over every irreducible component of V , it follows that] the degree of each of the ≥ 2 [cf. Proposition 2.5, (vi)] connected components of C c × C c η c over η c is equal to p — in contradiction to the fact that the degree of C c × C c η c over η c is ≤ p. This completes the proof of the claim. In particular, it follows that r E ≤ r E . Now recall that we have natural isomorphisms ∼ ω E /k (D E ) → ω (V ) log /S log | E ; ∼ ω E /k (D E ) → ω (V ) log /S log | E [cf. Proposition 2.5, (i), (ii), (iii), (iv), (v), (vi); the fact that the log-modiﬁcations (V ) log → (X ) log , (V ) log → (X ) log are unramiﬁed]. Moreover, it follows immedi- ately from the deﬁnitions that deg(ω E /k (D E )) = 2g E + r E , deg(ω E /k (D E )) = 2g E +r E . We thus conclude that deg(ω (V ) log /S log | C ) ≤ deg(ω (V ) log /S log | C ). On the other hand, the existence of the generically nonzero morphism of line bundles δ ∗ | C implies that deg(ω (V ) log /S log | C ) ≥ deg(ω (V ) log /S log | C ) = p · deg(ω (V ) log /S log | C ) ≥ p · deg(ω (V ) log /S log | C ) — which implies that deg(ω (V ) log /S log | C ) ≤ 0. Now if C is stable, then we have deg(ω (V ) log /S log | C ) = 2g E + r E > 0. We thus conclude that C is non- stable. But this implies that δ ∗ | C has at least one zero, so [cf. the above display of inequalities] we obtain that deg(ω (V ) log /S log | C ) < 0, in contradiction to the equality deg(ω (V ) log /S log | C ) = 0 if C is non-stable [cf. the proof of Proposition 2.5, (v)]. This completes the proof of assertion (ii). In light of assertion (ii), assertion (iii) follows immediately from Proposition 2.5, (viii). This completes the proof of assertion (iii). Next, we consider assertion (iv). First, let us observe that by Proposition 2.5, (ix), it follows that we may assume that split, base-ﬁeld-isomorphic log-modiﬁcations W log → Y log , Q log → X log , together with a morphism W log → Q log over Y log → X log , have been chosen so that the generic point of the unique irreducible compo- [1] nent C W of W that maps ﬁnitely to C maps into U Q . Then observe that there are precisely two mutually exclusive possibilities: 50 SHINICHI MOCHIZUKI [1] (c) some nonempty open subscheme of C maps quasi-ﬁnitely to U Q ; [1] (d) C maps to a closed point c of U Q . Moreover, by Proposition 2.5, (i), (ii), (vii), it follows immediately that (b) [as in the statement of assertion (iv)] ⇐⇒ (d). Thus, it suﬃces to show that (a) [as in the statement of assertion (iv)] ⇐⇒ (d). Note that it is immediate that (d) implies (a): Indeed, if we write c log for the log scheme obtained by equipping c with the restriction to c of the log structure of Q log , then we obtain an open injection π 1 (c log ) → G k log ; but this implies that the natural homomorphism Δ C → Δ X log factors through {1} = Ker(π 1 (c log ) → G k log ), hence that condition (a) is satisﬁed. Thus, it remains to show that (a) implies (d), or, equivalently, that condition (c) implies that condition (a) fails to hold. But this follows immediately from the observation that condition (c) implies that Δ C surjects onto an open subgroup of the decomposition group Δ E in Δ X log of some irreducible component E of Q. Here, we recall that the following well-known facts: if E is stable, then Δ E may be identiﬁed [1] def with the maximal pro-l quotient of π 1 (U E × k k), where we write U E = E U Q , which is inﬁnite; if E is not stable, then Δ E ∼ = Z l (1) [cf. Proposition 2.5, (v)], hence inﬁnite. This completes the proof of assertion (iv). Finally, we consider assertion (v). First, we observe that Δ C may be identiﬁed [1] def with the maximal pro-l quotient of π 1 (U C × k k), where we write U C = C U Y [and assume, for simplicity, that Y log is split]. In particular, an automorphism of U C is equal to the identity if and only if it induces the identity outer automorphism of Δ C [cf., e.g., [MT], Proposition 1.4, and its proof]. Note, moreover, that an automorphism of Y log stabilizes C if and only if it stabilizes the conjugacy class of Δ C [cf., e.g., [Mzk13], Proposition 1.2, (i)]. Thus, assertion (v) reduces to the [easily veriﬁed] assertion that the morphism Y η → X η is wildly ramiﬁed at C if and only if Gal(Y η /X η ) stabilizes and induces the identity on C. This completes the proof of assertion (v). Remark 2.6.1. Note that the content of Lemma 2.6, (ii), (iii), is reminiscent of the main results of [Tama2] [cf. also Corollary 2.11 below]. By comparison to Tamagawa’s “resolution of nonsingularities”, however, Lemma 2.6, (ii), (iii), assert a somewhat weaker conclusion, albeit for pro-p geometric fundamental groups, as opposed to proﬁnite geometric fundamental groups. Remark 2.6.2. The argument applied in the ﬁnal portion of the proof of Lemma 2.6, (ii), is reminiscent of the well-known classical argument that implies the nonex- istence of a Frobenius lifting for stable curves over the ring of Witt vectors of a ﬁnite ﬁeld. That is to say, if k is absolutely unramiﬁed, and Φ: X → X is an S-morphism that induces the Frobenius morphism between the respective spe- cial ﬁbers, then one obtains a contradiction as follows: Since Φ induces a morphism X η → X η , it follows immediately that Φ extends to a morphism of log stable curves Φ log : X log → X log . Although the derivative dΦ log : Φ ∗ (ω X log /S log ) → ω X log /S log TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 51 is ≡ 0 (mod p), one veriﬁes immediately [by an easy local calculation] that p 1 dΦ log is necessarily ≡ 0 (mod p) generically on each irreducible component of X. Since ω X log /S log is a line bundle of degree 2g X − 2, and Φ reduces to the Frobenius mor- phism between the special ﬁbers, the existence of dΦ log thus implies [by taking degrees] that p · (2g X − 2) ≤ 2g X − 2, i.e., that (p − 1)(2g X − 2) ≤ 0, in contradic- tion to the fact that g X ≥ 2. Remark 2.6.3. Note that it follows immediately from either of the conditions (a), (b) of Lemma 2.6, (ii), that Y is not k-smooth [i.e., “singular”]. Corollary 2.7. (Uniformly Toral Neighborhoods via Cyclic Coverings) Suppose that we are either in the situation of Lemma 2.6, (ii), (a) — which we shall refer to in the following as case (a) — or in the situation of Lemma 2.6, (ii), (b) — which we shall refer to in the following as case (b); suppose further, in case (b), that X is not smooth over k. Also, we suppose that we have been given a commutative diagram as in Lemma 2.6, (iii). Thus, in either case, we have an irreducible component C W of W [lying over an irreducible component C of Y ] [1] satisfying certain special properties, as in Lemma 2.6, (iii). Let y ∈ U W (O k ) (⊆ W η (k) = Y η (k)) be a point such that the image of y meets C W , and, moreover, y [1] maps to a point x ∈ U Q (O k ) (⊆ Q η (k) = X η (k)); C Q the irreducible component of Q that meets the image of x; def F y = W \(C W [1] U W ) ⊆ W ; def F x = Q\(C Q [1] U Q ) ⊆ Q [regarded as closed subsets of W , Q]; def U y = W \F y ⊆ W ; def U x = Q\F x ⊆ Q [so the image of y lies in U y ; the image of x lies in U x ]. Write g Y for the genus of Y η ; J η Y → η (respectively, J η X → η) for the Jacobian of Y η (respectively, X η ); J Y → S (respectively, J X → S) for the uniquely determined semi-abelian scheme over S that extends J η Y (respectively, J η X ); ι Yη : Y η → J η Y for the morphism that sends a T -valued point ζ [where T is a k-scheme] of Y η , regarded as a divisor on Y η × k T , to the point of J η Y determined by the degree zero divisor ζ − (y| T ). In case (a), let σ ∈ Gal(Y η /X η ) be a generator of Gal(Y η /X η ); J η ⊆ J η Y the image abelian scheme of the restriction to η of the endomorphism (1 − σ) : J Y → J Y ; J → S the uniquely determined semi-abelian scheme over S that extends J η [which exists, for instance, by [BLR], §7.4, Lemma 2]; κ : J Y → J the [dominant] morphism induced by (1 − σ). In case (b), let J → S be the semi- abelian scheme J X → S; κ : J Y → J the [dominant] morphism induced by the covering Y η → X η . Write β η : Y η × k . . . × k Y η → J η Y → J η 52 SHINICHI MOCHIZUKI [where the product is of g Y copies of Y η ] for the composite of the morphism given def by adding g Y copies of ι Yη with the morphism κ η = κ| η . m ) for the formal group over S given by com- (i) Write J (respectively, G pleting J (respectively, the multiplicative group (G m ) S over S) at the origin. Then there exists an exact sequence 0 → J → J → J → 0 ∼ of [formally smooth] formal groups over S, together with an isomorphism J → G m ∼ of formal groups over S. In the following, let us ﬁx such an isomorphism J → G m and identify J with its image in J. (ii) The morphisms ι Yη , β η extend uniquely to morphisms ι Y : U y → J Y ; β : U y × k . . . × k U y → J [where the product is of g Y copies of U y ], respectively; the morphism W → Q restricts to a morphism U y → U x . (iii) Suppose that k is an MLF [or, equivalently, that k is ﬁnite]. Then there exists a positive integer M — which, in fact, may be taken to be 1 in case (b) — such that the following condition holds: Let k • ⊆ k be a ﬁnite extension of k with ring of integers O k • , maximal ideal m k • ⊆ O k • . Write I k • for the image in J(O k • ) via β [cf. (ii)] of the product of g Y copies of U y (O k • ). Then M · I k • lies in the k ) ⊆ J(O k ). Write I k ⊆ J(O k ) for the subgroup determined by subgroup J(O • • • • M · I k • ; ∼ ∼ N k • ⊆ (O k × • ) ⊗ Q p → G m (O k • ) ⊗ Q p → J (O k • ) ⊗ Q p for the image of the intersection J (O k • ) I k • k )) (⊆ J(O • in J (O k • )⊗Q p . Then as k • ⊆ k varies over the ﬁnite extensions of k, the subgroups N k • determine a uniformly toral neighborhood of Gal(k/k) [cf. [Mzk15], Def- inition 3.6, (i), (ii)]. Proof. First, we consider assertion (i). Recall from the well-known theory of Néron models of Jacobians [cf., e.g., [BLR], §9.2, Example 8] that the torus portion of the special ﬁber of J Y (respectively, J X ) is [in the notation of Proposition 2.5, (vi)] of rank lp-rk(Y ) (respectively, lp-rk(X)). In particular, the torus portion of the special ﬁber of J is of rank lp-rk(Y ) − lp-rk(X) in case (a), and of rank lp-rk(X) in case (b). Thus, in case (a), the fact that the morphism Y η → X η is loopifying implies that the torus portion of the special ﬁber of J is of positive rank; in case (b), since X is not k-smooth, it follows from the loop-ampleness assumption of Lemma 2.6 that lp-rk(X) > 0, hence that the torus portion of the special ﬁber of J is of positive rank. Now the existence of an exact sequence as in assertion (i) follows from the well-known theory of degeneration of abelian varieties [cf., e.g., [FC], Chapter III, Corollary 7.3]. This completes the proof of assertion (i). TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 53 Next, we consider assertion (ii). The existence of the unique extension of ι Yη follows immediately from Proposition 2.5, (x); the existence of the unique extension of β η follows immediately from the existence of this unique extension of ι Yη [together with the existence of the homomorphism of semi-abelian schemes κ : J Y → J]. In case (b), the existence of the morphism U y → U x follows immediately from the deﬁnitions. In case (a), if the morphism U y → U x fails to exist, then there exists a [1] closed point w ∈ W that maps to a closed point q ∈ Q such that w ∈ U y ⊆ U W , but [1] q ∈ U Q . On the other hand, since there exists an irreducible Zariski neighborhood of w in W [cf. the simple structure of the monoid N], it follows from the fact that C W maps ﬁnitely to C Q [in case (a)], that W → Q is quasi-ﬁnite in a Zariski neighborhood of w. Thus, if we write R w , R q for the respective strict henselizations of W , Q at [some choice of k-valued points lifting] w, q, then R w , R q are normal, and theorem]. In particular, the natural inclusion R q → R w is ﬁnite [cf. Zariski’s main × K q = R q × [where “×” if we write K q for the quotient ﬁeld of R q , then we have R w denotes the subgroup of units], so the morphism W log → Q log induces an injection on the groupiﬁcations of the stalks of the characteristic sheaves at [some choice [1] of k-valued points lifting] w, q — in contradiction to the fact that w ∈ U W , but [1] q ∈ U Q . This completes the proof of assertion (ii). Finally, we consider assertion (iii). First, we deﬁne the number M as follows: In case (a), the endomorphism (1−σ) : J Y → J Y admits a factorization θ◦κ, where θ : J → J Y is a “closed immersion up to isogeny” [cf., e.g., the situation discussed in [BLR], §7.5, Proposition 3, (b)] — i.e., there exists a morphism θ : J Y → J such that θ ◦ θ : J → J is multiplication by some positive integer M θ on J; then def def we take M = M θ . In case (b), we take M = 1. In the following, if G is a group scheme or formal group over O k , and r ≥ 1 is an integer, then let us write G m r (O k • ) ⊆ G(O k • ) for the subgroup of elements that are congruent to the identity modulo m rk · O k • . Next, let us make the following observation: (1) We have M · I k • ⊆ J m (O k • ) ⊆ J(O k • ). Indeed, in case (a), we reason as follows: It suﬃces to show that M ·κ(ι Y (U y (O k • ))) ⊆ J m (O k • ). Since, moreover, the endomorphism (1 − σ) : J Y → J Y admits a factor- ization θ ◦ κ, where, for some morphism θ : J Y → J, θ ◦ θ is equal to multiplication by M , it suﬃces to show that Y (O k • ) (1 − σ)(ι Y (U y (O k • ))) ⊆ J m [since applying θ to this inclusion yields the desired inclusion M ·κ(ι Y (U y (O k • ))) ⊆ J m (O k • )]. On the other hand, since Y η → X η is wildly ramiﬁed at C W , it follows that σ acts as the identity on C W , hence that the composite morphism (1 − σ) ◦ ι Y : U y → J Y induces a morphism on special ﬁbers U y × O k k → J Y × O k k that is constant [with image lying in the image of the identity section of J Y × O k k]. But this implies Y (O k • ). This completes the proof of observation (1) that (1 − σ)(ι Y (U y (O k • ))) ⊆ J m in case (a). In a similar [but slightly simpler] vein, in case (b), it suﬃces to observe 54 SHINICHI MOCHIZUKI that the morphism κ ◦ ι Y : U y → J admits a factorization U y → U x → J X , where U y → U x is the morphism of assertion (ii), and U x → J X is the “analogue of ι Y ” for the point x of X η (k) [cf. Proposition 2.5, (x)]. That is to say, the fact that C W [1] maps to x ∈ U Q (O k ) implies [by applying this factorization] that the morphism κ ◦ ι Y : U y → J induces a morphism on special ﬁbers U y × O k k → J × O k k that is constant [with image lying in the image of the identity section of J × O k k]. This completes the proof of observation (1) in case (b). Next, let us make the following observation: (2) There exists a positive integer r which is independent of k • such that M · I k • ⊇ J m r (O k • ). Indeed, since κ is clearly dominant, it follows immediately that the composite of the morphism β : U y × k . . . × k U y → J with the morphism J → J given by multiplication by M is dominant, hence, in particular, generically smooth [since k is of characteristic zero]. Thus, [since M · I k • is a group!] observation (2) follows immediately from the “positive slope version of Hensel’s lemma” given in Lemma (O k • ) = J m (O k • ) J (O k • ) [cf. assertion (i)], we conclude that 2.1. Now since J m J m r (O k • ) ⊆ I k • J (O k • ) ⊆ J m (O k • ) [cf. the inclusions of observations (1), (2)], so assertion (iii) follows essentially formally [cf. [Mzk15], Deﬁnition 3.6, (i), (ii)]. This completes the proof of assertion (iii). Remark 2.7.1. Note that in the situation of case (b), if f is a rational function on X whose value at x lies in O k × , then the values ∈ O k × • of f at points of U y (O k • ) [cf. the notation of Corollary 2.7, (iii)] determine a uniformly toral neighborhood. It was precisely this observation that motivated the author to develop the theory of the present §2. Deﬁnition 2.8. k. Let k be a ﬁeld of characteristic zero, k an algebraic closure of (i) Suppose that k is equipped with a topology. Let X be a smooth, geo- metrically connected curve over k. Then we shall say that a subset Ξ ⊆ X(k) is Galois-dense if, for every ﬁnite extension ﬁeld k ⊆ k of k, Ξ X(k ) is dense in X(k ) [i.e., relative to the topology induced on X(k ) by k]. (ii) We shall refer to as a pro-curve U over k [cf. the terminology of [Mzk3]] any k-scheme U that may be written as a projective limit of smooth, geometrically connected curves over k in which the transition morphisms are birational. Let U be a pro-curve over k. Then it makes sense to speak of the function ﬁeld k(U ) of U . Write X for the smooth, proper, geometrically connected curve over k determined by the function ﬁeld k(U ). Then one veriﬁes immediately that U is completely determined up to unique isomorphism by k(U ), together with some Gal(k/k)-stable subset Ξ ⊆ X(k) — i.e., roughly speaking, “U is obtained by removing Ξ from X”. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 55 If k is equipped with a topology, then we shall say that U is co-Galois-dense if the corresponding Gal(k/k)-stable subset Ξ ⊆ X(k) is Galois-dense. Remark 2.8.1. Suppose, in the notation of Deﬁnition 2.8, that k is an MLF [and that k is equipped with the p-adic topology]. Let X be a smooth, proper, geometrically connected curve over k, with function ﬁeld k(X). Then Spec(k(X)) is a co-Galois-dense pro-curve over k. Suppose that X = X 0 × k 0 k, where k 0 ⊆ k is a number ﬁeld, and X 0 is a smooth, proper, geometrically connected curve over k 0 , with function ﬁeld k(X 0 ). Then Spec(k(X 0 ) ⊗ k 0 k) [where we note that the ring k(X 0 ) ⊗ k 0 k is not a ﬁeld!] also forms an example of a co-Galois-dense pro-curve over k. Remark 2.8.2. Let k be a ﬁeld of characteristic zero. (i) Let us say that a pro-curve U over k is of unit type if there exists a connected ﬁnite étale covering of U that admits a nonconstant unit. Thus, any hyperbolic curve U over k for which there exists a connected ﬁnite étale covering V → U such that V admits a dominant k-morphism V → P , where P is the projective line minus three points over k, is of unit type. That is to say, the hyperbolic curves considered in [Mzk15], Remark 3.8.1 — i.e., the sort of hyperbolic curves that motivated the author to prove [Mzk15], Corollary 3.8, (g) — are necessarily of unit type. (ii) Suppose that k is an MLF of residue characteristic p, whose ring of integers we denote by O k . Let n ≥ 1 be an integer; η ∈ O k /(p n ). Then observe that the set E of elements of O k that are ≡ η (mod p n ) is of uncountable cardinality. In particular, it follows that the subﬁeld of k generated over Q by E is of uncountable — hence, in particular, inﬁnite — transcendence degree over Q. (iii) Let k be as in (ii); X 0 a proper hyperbolic curve over k 0 , where k 0 ⊆ k is a ﬁnitely generated extension of Q; k 1 ⊆ k a ﬁnitely generated extension of k 0 ; r ≥ 1 an integer. Then recall from [MT], Corollary 5.7, that any curve U 1 obtained by def removing from X 1 = X 0 × k 0 k 1 a set of r “generic points” ∈ X 1 (k 1 ) = X 0 (k 1 ) — i.e., r points which determine a dominant morphism from Spec(k 1 ) to the product of r copies of X 0 over k 0 — is not of unit-type. In particular, it follows immediately from (ii) that: There exist co-Galois-dense pro-curves U over k which are not of unit type. For more on the signiﬁcance of this fact, we refer to Remark 2.10.1 below. Remark 2.8.3. Suppose that we are in the situation of Deﬁnition 2.8, (i). Let Y → X be a connected ﬁnite étale covering. Then one veriﬁes immediately, by applying “Krasner’s Lemma” [cf. [Kobl], pp. 69-70], that the inverse image Ξ| Y ⊆ Y (k) of a Galois-dense subset Ξ ⊆ X(k) is itself Galois-dense. 56 SHINICHI MOCHIZUKI Corollary 2.9. (Point-theoreticity Implies Geometricity) For i = 1, 2, let k i be an MLF of residue characteristic p i ; k i an algebraic closure of k i ; Σ i a set of primes of cardinality ≥ 2 such that p i ∈ Σ i ; X i a hyperbolic curve over k i ; X i the smooth, proper, geometrically connected curve over k i determined by the function ﬁeld of X i ; Ξ i ⊆ X i (k i ) a Galois-dense subset. Write “π 1 (−)” for the étale fundamental group of a connected scheme, relative to an appropriate choice of basepoint; Δ X i for the maximal pro-Σ i quotient of π 1 (X i × k i k i ); Π X i for the quotient of π 1 (X i ) by the kernel of the natural surjection π 1 (X i × k i k i ) Δ X i . Let ∼ α : Π X 1 → Π X 2 be an isomorphism of proﬁnite groups such that a closed subgroup of Π X 1 is a decomposition group of a point ∈ Ξ 1 if and only if it corresponds, relative to α, to a decomposition group in Π X 2 of a point ∈ Ξ 2 . Then p 1 = p 2 , Σ 1 = Σ 2 , and α is ∼ geometric, i.e., arises from a unique isomorphism of schemes X 1 → X 2 . Proof. First, we observe that by [Mzk15], Theorem 2.14, (i), α induces iso- ∼ ∼ morphisms α Δ : Δ X 1 → Δ X 2 , α G : G 1 → G 2 [where, for i = 1, 2, we write def def G i = Gal(k i /k i )]; p 1 = p 2 [so we shall write p = p 1 = p 2 ]; Σ 1 = Σ 2 [so we shall def write Σ = Σ 1 = Σ 2 ]. Also, by [the portion concerning semi-graphs of] [Mzk15], Theorem 2.14, (i), it follows that α preserves the decomposition groups of cusps. Thus, by passing to corresponding open subgroups of Π X i [cf. Remark 2.8.3] and forming the quotient by the decomposition groups of cusps in Δ X i , we may assume, without loss of generality, that the X i are proper. Next, let l ∈ Σ be a prime = p. Then let us recall that by the well-known stable reduction criterion [cf., e.g., [BLR], §7.4, Theorem 6], X i has stable reduction over O k i if and only if, for some Z l - ab submodule M ⊆ Δ ab X i ⊗ Z l of the maximal pro-l abelian quotient Δ X i ⊗ Z l of Δ X i , the inertia subgroup of G i acts trivially on M , Δ ab X i ⊗ Z l /M . Thus, we may assume, without loss of generality, that, for i = 1, 2, X i admits a log stable model X i log over Spec(O k i ) log [where the last log structure is the log structure determined by the closed point]. Since, by [the portion concerning semi-graphs of] [Mzk15], Theorem 2.14, (i), it follows that α induces an isomorphism between the dual graphs of the special ﬁbers X i of the X i , hence that X 1 is loop-ample (respectively, singular) if and only if X 2 is. Thus, by replacing X i by a ﬁnite étale covering of X i arising from an open subgroup of Π X i , we may assume that X i is loop-ample [cf. §0] and singular [cf. Remark 2.6.3]. Now, to complete the proof of Corollary 2.9, it follows from [Mzk15], Corollary 3.8, (e), that it suﬃces to show that α G is uniformly toral. Next, let us suppose that, for i = 1, 2, we are given a ﬁnite étale covering Y i → X i of hyperbolic curves over k i with stable reduction over O k i arising from open subgroups of Π X i that correspond via α and are such that Gal(Y i /X i ) is cyclic of order a positive power of p [cf. Lemma 2.6, (ii)]. Let us write Y i log for the log stable model of Y i over (O k i ) log , Y i for the special ﬁber of Y i . By possibly replacing the k i by corresponding [relative to α] ﬁnite extensions of k i , we may assume that the Y i are split [cf. Proposition 2.5, (vi)]. Note that by [the portion concerning semi-graphs of] [Mzk15], Theorem 2.14, (i), it follows that Y 1 → X 1 is loopifying if and only if Y 2 → X 2 is. Thus, by Lemma 2.6, (iv), (v) [cf. also Proposition 2.5, (i), (ii)] (respectively, Lemma 2.6, (iv)), it follows that Y 1 → X 1 satisﬁes condition (a) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 57 (respectively, (b)) of Lemma 2.6, (ii), if and if Y 2 → X 2 does. For i = 1, 2, let C i be an irreducible component of Y i as in Corollary 2.7 [i.e., “C”]. By Lemma 2.6, (iv) [cf. also the portion concerning semi-graphs of [Mzk15], Theorem 2.14, (i)], we may assume that the C i are compatible with α. Thus, to complete the proof of Corollary 2.9, it suﬃces to construct uniformly toral neighborhoods [cf. Corollary 2.7, (iii)] that are compatible with α. Write Π Y i ⊆ Π X i , Δ Y i ⊆ Δ X i for the open subgroups determined by Y i ; T i Y , T i X for the maximal pro-p abelian quotients of Δ Y i , Δ X i . If we are in case (a) [cf. Corollary 2.7], then we choose generators σ i ∈ Gal(Y i /X i ) ∼ = Δ X i /Δ Y i that correspond via α and write T i for the intersection with T i Y of the image of the endomorphism (1 − σ i ) of T i Y ⊗ Q p , and κ T i : T i Y → T i for the morphism induced by (1 − σ i ). If we are in case (b) [cf. Corollary 2.7], then def we set T i = T i X ; write κ T i : T i Y → T i for the morphism induced by Y i → X i [i.e., by the inclusion Δ Y i → Δ X i ]. Thus, the formal group “J ” of Corollary 2.7, (i), corresponds to a G i -submodule T i ⊆ T i such that T i ∼ = Z p (1); by [Tate], Theorem 4 [cf. also [Mzk5], Proposition 1.2.1, (vi)], we may assume that these submodules T i are compatible with α. (l) (l) (l) Next, let us write Δ Y i Δ Y i for the maximal pro-l quotient of Δ Y i ; Δ C i ⊆ Δ Y i (l) for the decomposition group of C i in Δ Y i [well-deﬁned up to conjugation]. Thus, (l) Δ C i may be identiﬁed with the maximal pro-l quotient of π 1 (U C i × k i k i ), where [1] def U C i = C i U Y i [cf. the proof of Lemma 2.6, (iv)], k i is the residue ﬁeld of k i , (l) and k i is the algebraic closure of k i induced by k i . Since Δ C i is slim [cf., e.g., (l) [MT], Proposition 1.4], and the outer action of G i on Δ C i clearly factors through (l) the quotient G i Gal(k i /k i ), the resulting outer action of Gal(k i /k i ) on Δ C i determines, in a fashion that is compatible with α, an extension of proﬁnite groups (l) (l) 1 → Δ C i → Π C i → Gal(k i /k i ) → 1. In a similar vein, the outer action of G i (l) on Δ Y i factors through the maximal tamely ramiﬁed quotient G i G k log , hence i (l) [since Δ Y i is slim — cf., e.g., [MT], Proposition 1.4] determines, in a fashion that is compatible with α, a morphism of extensions of proﬁnite groups (l) −→ Π C i × Gal(k /k ) G k log i ⏐ i i ⏐ (l) −→ 1 −→ Δ C i ⏐ ⏐ 1 −→ Δ Y i (l) (l) Π Y i −→ G k log ⏐ i ⏐ −→ 1 −→ G k log −→ 1 i — in which the vertical morphisms are inclusions, and the vertical morphism on the right is the identity morphism; moreover, the images of the ﬁrst two vertical morphisms are equal to the respective decomposition groups of C i [well-deﬁned up to conjugation]. Next, let us observe that, by our assumption concerning decomposition groups of points ∈ Ξ i in the statement of Corollary 2.9, it follows that α determines ∼ a bijection Y 1 (k 1 , Ξ 1 ) → Y 2 (k 2 , Ξ 2 ), where we write Y i (k i , Ξ i ) ⊆ Y i (k i ) for the 58 SHINICHI MOCHIZUKI subset of points lying over points ∈ Ξ i . [Here, we recall that a point ∈ Y i (k i ) is uniquely determined by the conjugacy class of its decomposition group in Π Y i — cf., e.g., [Mzk3], Theorem C.] Now let us choose corresponding [i.e., via this bijection] points y i ∈ Y i (k i , Ξ i ) as our points “y” in the construction of the uniformly toral neighborhoods of Corollary 2.7, (iii). Here, we observe that [by our Galois- density assumption] we may assume that y i is compatible with C i , in the sense (l) that the image in the quotient Π Y i Π Y i of the decomposition group of y i in (l) Π Y i determines a subgroup of Π C i × Gal(k /k ) G k log which contains the kernel of i (l) (l) i i the surjection Π C i × Gal(k /k ) G k log Π C i . Note that this condition that y i be i i i “compatible with C i ” is manifestly “group-theoretic”, i.e., compatible with α [cf. the portion concerning semi-graphs of [Mzk15], Theorem 2.14, (i); [Mzk5], Proposition 1.2.1, (ii)]. Moreover, let us recall from the theory of §1 that this condition that y i be “compatible with C i ” is equivalent to the condition that the closure in Y i of y i intersect U C i [cf. Proposition 1.3, (x)]. Thus, by choosing any corresponding [i.e., via the bijection induced by α] points y i ∈ Y i (k i , Ξ i ) that are compatible with the C i , we may compute [directly from the decomposition groups of the y i , y i in Π Y i ] the “diﬀerence” of y i , y i in H 1 (G k i , T i Y ), as well as the image δ y i ,y i ∈ H 1 (G k i , T i ) of this diﬀerence via κ T i . On the other hand, let us recall the Kummer isomorphisms H 1 (G k i , T i ) ∼ = O k × i ⊗ Z p ; H 1 (G k i , T i ) ∼ = J i (k i ) ⊗ Z p [where J i is the “J” of Corollary 2.7, (iii) — cf., e.g., the “well-known general nonsense” reviewed in the proof of [Mzk14], Proposition 2.2, (i), for more details]. By applying these isomorphisms, we conclude that the subset of H 1 (G k i , T i ⊗ Q p ) ∼ = O k × i ⊗ Q p obtained by taking the image of the intersection in H 1 (G k i , T i ) with the image of H 1 (G k i , T i ) of the closure [cf. our Galois-density assumption, together with the evident p-adic continuity of the assignment y i → δ y i ,y i ] of the set obtained by adding g Y i [where g Y i is the genus of Y i ] elements of the form M i · δ y i ,y i [where M i is the “M ” of Corollary 2.7, (iii)] yields — from the point of view of Corollary 2.7, (iii) — a subset that coincides with the subset “N k • ” [when “k • ” is taken to be k i ] constructed in Corollary 2.7, (iii). Thus, by allowing the “k i ” to vary over arbitrary corresponding ﬁnite extensions ⊆ k i of k i , we obtain uniform toral neighborhoods of the G i that are compatible with α. But this implies that α G is uniformly toral, hence completes the proof of Corollary 2.9. Remark 2.9.1. Corollary 2.9 may be regarded as a generalization of the [MLF portion of] [Mzk14], Corollary 2.2, to the case of pro-Σ [where Σ is of cardinality ≥ 2 and contains the residue characteristic — that is to say, Σ is not necessarily the set of all primes] geometric fundamental groups of not necessarily aﬃne hyperbolic curves. From this point of view, it is interesting to note that in the theory of the TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 59 present §2, Lemma 2.6, which, as is discussed in Remark 2.6.2, is reminiscent of a classical argument on the “nonexistence of Frobenius liftings”, takes the place of Lemma 4.7 of [Tama1], which is applied in [Mzk14], Corollary 2.1, to reconstruct the additive structure of the ﬁelds involved. In this context, we observe that the appearance of “Frobenius endomorphisms” in Remark 2.6.2 is interesting in light of the discussion of “hidden endomorphisms” in the Introduction, in which “Frobenius endomorphisms” also appear. Remark 2.9.2. One way to think of Corollary 2.9 is as the statement that: The “Section Conjecture” over MLF’s implies the “absolute isomorphism version of the Grothendieck Conjecture” over MLF’s. Here, we recall that in the notation of Corollary 2.9, the “Section Conjecture” over MLF’s amounts to the assertion that every closed subgroup of Π X i that maps isomorphically to an open subgroup of Gal(k i /k i ) is the decomposition group asso- ciated to a closed point of X i . In fact, in order to apply Corollary 2.9, a “relatively weak version of the Section Conjecture” is suﬃcient — cf. the point of view of [Mzk8]. Remark 2.9.3. The issue of verifying the “point-theoreticity hypothesis” of Corol- lary 2.9 [i.e., the hypothesis concerning the preservation of decomposition groups of closed points] may be thought of as consisting of two steps, as follows: (a) First, one must show the J-geometricity [cf. [Mzk3], Deﬁnition 4.3] of the image via α of a decomposition group D ξ ⊆ Π X 1 of a closed point ξ ∈ X 1 (k 1 ). Once one shows this J-geometricity for all ﬁnite étale coverings of X 2 arising from open subgroups of Π X 2 , one concludes [cf. the arguments of [Mzk3], §7, §8] that there exist rational points of a certain tower of coverings of X 2 determined by α(D ξ ) ⊆ Π X 2 over tame extensions of k 2 . (b) Finally, one must show that these rational points over tame extensions of k 2 necessarily converge — an issue that the author typically refers to by the term “tame convergence”. At the time of writing, it is not clear to the author how to complete either of these two steps. On the other hand, in the “birational” — or, more generally, the “co-Galois-dense” — case, one has Corollary 2.10 [given below]. Remark 2.9.4. (i) By contrast to the quite substantial diﬃculty [discussed in Remark 2.9.3] of verifying “point-theoreticity” for hyperbolic curves over MLF’s, in the case of hyperbolic curves over ﬁnite ﬁelds, there is a [relatively simple] “group-theoretic” algorithm for reconstructing the decomposition groups of closed points, which follows essentially from the theory of [Tama1] [cf. [Tama1], Corollary 2.10, Proposition 3.8]. Such an algorithm is discussed in [Mzk14], Remark 10, although the argument given there is somewhat sketchy and a bit misleading. A more detailed presentation may be found in [SdTm], Corollary 1.25. 60 SHINICHI MOCHIZUKI (ii) A more concise version of this argument, along the lines of [Mzk14], Remark 10, may be given as follows: Let X be a proper [for simplicity] hyperbolic curve over a ﬁnite ﬁeld k, with algebraic closure k; Σ a set of prime numbers that contains a prime that is invertible in k; π 1 (X × k k) Δ X the maximal pro-Σ quotient of the étale fundamental group π 1 (X × k k) of X × k k; π 1 (X) Π X the corresponding def quotient of the étale fundamental group π 1 (X) of X; Π X G k = Gal(k/k) the natural quotient. Then it suﬃces to give a “group-theoretic” characterization of the quasi-sections D ⊆ Π X [i.e., closed subgroups that map isomorphically onto an open subgroup of G k ] which are decomposition groups of closed points of X. Write → X X for the pro-ﬁnite étale covering corresponding to Π X . If E ⊆ Π X is a closed subgroup whose image in G k is open, then let us write k E for the ﬁnite extension ﬁeld of k determined by this image. If J ⊆ Π X is an open subgroup, then let us write def X J → X for the covering determined by J and J Δ = J Δ X . If J ⊆ Π X is an open subgroup such that J Δ is a characteristic subgroup of Δ X , then we shall say that J is geometrically characteristic. Now let J ⊆ Π X be a geometrically characteristic open subgroup. Let us refer to as a descent-group for J any open subgroup H ⊆ Π X such that J ⊆ H, J Δ = H Δ . Thus, a descent-group H for J may be thought of as an intermediate covering X J → X H → X such that X H × k H k J ∼ = X J . Write X J (k J ) ﬂd-def ⊆ X J (k J ) for the subset of k J -valued points of X J that do not arise from points ∈ X H (k H ) for any descent-group H = J for J — i.e., the k J -valued points whose ﬁeld of deﬁnition is k J with respect to all possible “descended forms” of X J . [That is to say, this deﬁnition of “ﬂd-def” diﬀers slightly from the deﬁnition of “ﬂd-def” in [Mzk14], that maps to x ∈ X J (k J ), and we Remark 10.] Thus, if x is a closed point of X ⊆ Π X for the stabilizer in Π X [i.e., “decomposition group”] of x , then it write D x def · J Δ (⊇ J) [so H x is a tautology that x maps to a point ∈ X H x (k H x ) for H x = D x forms a descent-group for J]; in particular, it follows immediately that: x ∈ X J (k J ) ﬂd-def ⇐⇒ D ⊆ J x ⇐⇒ H x = J. Now it follows immediately from this characterization of “ﬂd-def” that if J 1 ⊆ J 2 ⊆ Π X are geometrically characteristic open subgroups such that k J 1 = k J 2 , then the natural map X J 1 (k J 1 ) → X J 2 (k J 2 ) induces a map X J 1 (k J 1 ) ﬂd-def → X J 2 (k J 2 ) ﬂd-def . Moreover, these considerations allow one to conclude [cf. the theory of [Tama1]] that: A quasi-section D ⊆ Π X is a decomposition group of a closed point of X if and and only if, for every geometrically characteristic open subgroup J ⊆ Π X such that D · J Δ = J, it holds that X J (k J ) ﬂd-def = ∅. Thus, to render this characterization of decomposition groups “group-theoretic”, it suﬃces to give a “group-theoretic” criterion for the condition that X J (k J ) ﬂd-def = ∅. In [Tama1], the Lefschetz trace formula is applied to compute the cardinality of TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 61 X J (k J ). On the other hand, if we use the notation “| − |” to denote the cardinality of a ﬁnite set, then one veriﬁes immediately that |X H (k H ) ﬂd-def | |X J (k J )| = H — where H ⊇ J ranges over the descent-groups for J. In particular, by applying induction on [Π X : J], one concludes immediately from the above formula that |X J (k J ) ﬂd-def | may be computed from the |X H (k H )|, as H ranges over the descent- groups for J [while |X H (k H )| may be computed, as in [Tama1], from the Lefschetz trace formula]. This yields the desired “group-theoretic” characterization of the decomposition groups of Π X . Corollary 2.10. (Geometricity of Absolute Isomorphisms for Co- Galois-dense Pro-curves) For i = 1, 2, let k i be an MLF of residue charac- teristic p i ; k i an algebraic closure of k i ; Σ i a set of primes of cardinality ≥ 2 such that p i ∈ Σ i ; U i a co-Galois-dense pro-curve over k i . Write “π 1 (−)” for the étale fundamental group of a connected scheme, relative to an appropriate choice of basepoint; Δ U i for the maximal pro-Σ i quotient of π 1 (U i × k i k i ); Π U i for the quotient of π 1 (U i ) by the kernel of the natural surjection π 1 (U i × k i k i ) Δ U i . Let ∼ α : Π U 1 → Π U 2 be an isomorphism of proﬁnite groups. Then Δ U i , Π U i are slim; p 1 = p 2 ; Σ 1 = Σ 2 ; α is geometric, i.e., arises from a unique isomorphism of schemes ∼ U 1 → U 2 . Proof. First, we observe that Σ i may be characterized as the set of primes l such that Π U i has l-cohomological dimension > 2. Thus, Σ 1 = Σ 2 . Let us write def Σ = Σ 1 = Σ 2 ; X i for the smooth, proper, geometrically connected curve over k i determined by U i ; Δ X i for the maximal pro-Σ quotient of π 1 (X i × k i k i ); Π X i for the quotient of π 1 (X i ) by Ker(π 1 (X i × k i k i ) Δ X i ). Thus, U i determines some Galois-dense subset Ξ i ⊆ X i (k i ). Since Δ U i , Π U i may be written as inverse limits of surjections of slim proﬁnite groups [cf., e.g., [Mzk15], Proposition 2.3], it follows that Δ U i , Π U i are slim. Since the kernel of the natural surjection Π U i Π X i is topologically generated by the inertia groups of points ∈ Ξ i , and these inertia Σ (1) [where the “(1)” denotes a Tate twist; we write groups are isomorphic to Z Σ for the maximal pro-Σ quotient of Z] in a fashion that is compatible with the Z def conjugation action of some open subgroup of G i = Gal(k i /k i ), it follows that we obtain an isomorphism ∼ ab-t Π ab-t U i → Π X i on torsion-free abelianizations. In particular, it follows [in light of our assumptions on Σ i ] that, in the notation of [Mzk15], Theorem 2.6, sup p ,p ∈Σ {δ p 1 (Π U i ) − δ p 1 (Π U i )} = sup p ,p ∈Σ = sup p ,p ∈Σ {δ p 1 (Π X i ) − δ p 1 (Π X i )} {δ p 1 (G i ) − δ p 1 (G i )} = [k i : Q p i ] 62 SHINICHI MOCHIZUKI — cf. [Mzk15], Theorem 2.6, (ii). In particular, by applying this chain of equalities to arbitrary open subgroups of Π U i , we conclude that α induces an isomorphism ∼ α G : G 1 → G 2 . Moreover, by [Mzk5], Proposition 1.2.1, (i), (vi), the existence of def α G implies that p 1 = p 2 [so we set p = p 1 = p 2 ], and α G is compatible with the respective cyclotomic characters. Let M be a proﬁnite abelian group equipped with a continuous H-action, for H ⊆ G i [where i ∈ {1, 2}] an open subgroup. Then let us write M Q (M ) for the quotient of M by the closed subgroup generated by the quasi-toral subgroups of M [i.e., closed subgroups isomorphic as J-modules, for J ⊆ H an open subgroup, to Z l (1) for some prime l]; M Q (M ) Q(M ) for the maximal torsion-free quotient of Q (M ). Also, if M is topologically ﬁnitely generated, then let us write M T (M ) for the maximal torsion-free quasi-trivial quotient [i.e., maximal torsion-free quotient on which H acts through a ﬁnite quotient]. Then one veriﬁes immediately that the assignments M → Q(M ), M → T (M ) are functorial. Moreover, it follows from the observations of the preceding paragraph that the natural surjection Δ U i Δ ab-t Δ X i determines a surjection on torsion-free abelianizations Δ ab-t U i X i that ∼ ab-t ab-t induces an isomorphism Q(Δ U i ) → Q(Δ X i ). Thus, it follows from “Poincaré ∼ ab-t duality” [i.e., the isomorphism Δ ab-t X i → Hom(Δ X i , Z(1)) determined by the cup- product on the étale cohomology of X] that ab-t 2g X i = dim Q l (Q(Δ ab-t X i ) ⊗ Q l ) + dim Q l (T (Δ X i ) ⊗ Q l ) ab-t = dim Q l (Q(Δ ab-t X i ) ⊗ Q l ) + dim Q l (T (Q(Δ X i )) ⊗ Q l ) ab-t = dim Q l (Q(Δ ab-t U i ) ⊗ Q l ) + dim Q l (T (Q(Δ U i )) ⊗ Q l ) — where g X i is the genus of X i , and l ∈ Σ. Thus, we conclude that g X 1 = g X 2 . In particular, by passing to corresponding [i.e., via α] open subgroups of the Π U i , we may assume that g X 1 = g X 2 ≥ 2. Next, by applying this equality “g X 1 = g X 2 ” to corresponding [i.e., via α] open subgroups of the Π U i , it follows from the Hurwitz formula that the condition on a pair of open subgroups J i ⊆ H i ⊆ Δ U i that “the covering between J i and H i be cyclic of order a power of a prime number and totally ramiﬁed at precisely one closed point but unramiﬁed elsewhere” is preserved by α. Thus, it follows formally [cf., e.g., the latter portion of the proof of [Mzk5], Lemma 1.3.9] that α preserves the inertia groups of points ∈ Ξ i . Moreover, by considering the conjugation action of Π U i on these inertia groups, we conclude that α preserves the decomposition groups ∼ ⊆ Π U i of points ∈ Ξ i . Thus, in summary, α induces an isomorphism Π X 1 → Π X 2 that preserves the decomposition groups ⊆ Π X i of points ∈ Ξ i ; in particular, by ∼ applying Corollary 2.9 to this isomorphism Π X 1 → Π X 2 , we obtain an isomorphism ∼ of schemes U 1 → U 2 , as desired. This completes the proof of Corollary 2.10. Remark 2.10.1. Thus, by contrast to the results of [Mzk14], Corollary 2.3, or [Mzk15], Corollary 3.8, (g) [cf. [Mzk15], Remark 3.8.1] — or, indeed, Corollary 1.11, (iii) of the present paper — Corollary 2.10 constitutes the ﬁrst “absolute isomorphism version of the Grothendieck Conjecture over MLF’s” known to the author that does not rely on the use of Belyi maps. One aspect of this independence of the theory of Belyi maps may be seen in the fact that Corollary 2.10 may be TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 63 applied to pro-curves which are not of unit type [cf. Remark 2.8.2, (i), (iii)]. Another aspect of this independence of the theory of Belyi maps may be seen in the fact that Corollary 2.10 involves geometrically pro-Σ arithmetic fundamental groups for Σ which are not necessarily equal to the set of all prime numbers. Finally, we observe that the techniques developed in the present §2 allow one to give a more pedestrian treatment of the [somewhat sketchy] treatment given in [Mzk9] [cf. the veriﬁcation of “observation (iv)” given in the proof of [Mzk9], Corollary 3.11, as well as Remark 2.11.1 below] of the fact that “cusps always appear as images of nodes”. Corollary 2.11. (Cusps as Images of Nodes) Let k be a complete dis- cretely valued ﬁeld of characteristic zero, with perfect residue ﬁeld k of char- def acteristic p > 0 and ring of integers O K ; η = Spec(k); S log the log scheme obtained def by equipping S = Spec(O K ) with the log structure determined by the closed point def S = Spec(k) of S; X log → S log a stable log curve over S log such that the un- derlying scheme of the generic ﬁber X log × S η is smooth; ξ ∈ X(S) a cusp of the stable log curve X log ; ξ ∈ X(S) the restriction of ξ to the special ﬁber X of X. In the following, we shall denote restrictions to η by means of a subscript η; also we shall often identify ξ with its image in X. Then, after possibly replacing k by a ﬁnite extension of k, there exists a morphism of stable log curves over S log φ log : Y log → X log such that the following properties are satisﬁed: log (a) the restriction φ log → X η log is a ﬁnite log étale Galois covering; η : Y η (b) ξ is the image of a node of the special ﬁber Y of Y ; (c) ξ is the image of an irreducible component of Y . If, moreover, X is sturdy, loop-ample, and singular, then φ η : Y η → X η may be taken to be ﬁnite étale of degree p. Proof. By replacing k by an appropriate subﬁeld of k, one veriﬁes immediately that we may assume that k is of countable cardinality, hence that k satisﬁes the hypotheses of the discussion preceding Lemma 2.6. After possibly replacing k by a ﬁnite extension of k and X η log by a ﬁnite log étale Galois covering of X η log [which, in fact, may be taken to be of degree a power of p · l, where l is a prime = p], we may assume that X is sturdy [cf. §0], loop-ample [cf. §0], singular [cf. Remark 2.6.3], and split. Next, let us recall from the well-known theory of pointed stable curves [cf. [Knud]] that if we write V log → S log for the stable log curve obtained by forgetting the cusps of X log [so V η = X η ], then it follows immediately from the fact that X is sturdy that V = X. Thus, by Lemma 2.6, (i) [cf. also the way in which Lemma 2.6, (i), is applied in the proof of Lemma 2.6, (ii)], it follows that, after 64 SHINICHI MOCHIZUKI possibly replacing k by a ﬁnite extension of k, there exists a morphism of stable log curves over S log φ log : Y log → X log such that if we write φ for the morphism of schemes underlying φ log , then φ η : Y η → X η is a ﬁnite étale Galois covering of degree p that is wildly ramiﬁed over the irreducible component C of X containing ξ. Next, let us suppose that the property (c) is satisﬁed. Thus, there exists an irreducible component E of Y that maps to ξ. Next, let us observe that there exists an irreducible component D of Y that maps ﬁnitely to C and meets the connected component F of the ﬁber φ −1 (ξ) that contains E [so D is not contained in F ]. In particular, it follows that there exists a chain of irreducible components E 1 = E, E 2 , . . . , E n [where n ≥ 1 is an integer] of Y joining E to D such that each E j ⊆ F [for j = 1, . . . , n]. Thus, E n meets D at some node of Y that maps to ξ. That is to say, property (b) is satisﬁed. Thus, to complete the proof of Corollary 2.11, it suﬃces to verify property (c). Now suppose that property (c) fails to hold. Then φ is ﬁnite over some neigh- borhood of ξ. Since φ η is wildly ramiﬁed over C, it follows that there exists a nontrivial element σ ∈ Gal(Y η log /X η log ) that ﬁxes and acts as the identity on some irreducible component D of Y that maps ﬁnitely to C. After possibly replacing k by a ﬁnite extension of k, it follows from the ﬁniteness of φ over some neighborhood of ξ that we may assume that there exists a cusp ζ ∈ Y (S) of Y log lying over ξ such that the restriction ζ of ζ to Y lies in D. But then the distinct [since σ is nontrivial, and φ η is étale] cusps ζ, ζ σ of Y log have identical restrictions ζ, ζ σ to Y — in contradiction to the deﬁnition of a “stable log curve” [i.e., of a “pointed stable curve”]. This completes the proof of property (c) and hence of Corollary 2.11. Remark 2.11.1. (i) The statement of [Mzk9], Corollary 3.11, concerns smooth log curves over an MLF, but in fact, the same proof as the proof given in [Mzk9] for [Mzk9], Corollary 3.11, may be applied to smooth log curves over an arbitrary mixed characteristic complete discretely valued ﬁeld. Here, we note that by passing to an appropriate extension, this discretely valued ﬁeld may be assumed to have a perfect residue ﬁeld, as in Corollary 2.11. In particular, Corollary 2.11 may be applied to [the portion corresponding to “observation (iv)” in loc. cit. of] the proof of such a generalization of [Mzk9], Corollary 3.11, for more general ﬁelds. (ii) In the discussion of the “pro-Σ version” of [Mzk9], Corollary 3.11, in [Mzk9], Remark 3.11.1, one should assume that p α , p β ∈ Σ. In fact, this assumption is, in some sense, implicit in the phraseology that appears in the ﬁrst two lines of [Mzk9], Remark 3.11.1, but it should have been stated explicitly. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 65 Section 3: Elliptic and Belyi Cuspidalizations The sort of preservation of decomposition groups of closed points that is re- quired in the hypothesis of Corollary 2.9 is shown [for certain types of hyperbolic curves] in the case of proﬁnite geometric fundamental groups in [Mzk8], Corollary 3.2. On the other hand, at the time of writing, the author does not know of any such results in the case of pro-Σ geometric fundamental groups, when Σ is not equal to the set of all primes. Nevertheless, in the present §3, we observe that the tech- niques of [Mzk8], §2, concerning the preservation of decomposition groups of torsion points of elliptic curves do indeed hold for fairly general pro-Σ geometric fundamen- tal groups [cf. Corollaries 3.3, 3.4]. Moreover, we observe that these techniques — which may be applied not only to [hyperbolic orbicurves related to] elliptic curves, but also, in the proﬁnite case, to [hyperbolic orbicurves related to] tripods [i.e., hyperbolic curves of type (0, 3) — cf. [Mzk15], §0], via the use of Belyi maps — allow one to recover not only the decomposition groups of [certain] closed points, but also the resulting “cuspidalizations” [i.e., the arithmetic fundamental groups of open subschemes obtained by removing such closed points] — cf. Corollaries 3.7, 3.8. Let X be a hyperbolic orbicurve over a ﬁeld k of characteristic zero; k an algebraic closure of k. We shall denote the base-change operation “× k k” by means of a subscript k. Thus, we have an exact sequence of fundamental groups 1 → π 1 (X k ) → π 1 (X) → Gal(k/k) → 1. Deﬁnition 3.1. Let π 1 (X) Π be a quotient of proﬁnite groups. Write Δ ⊆ Π for the image of π 1 (X k ) in Π. Then we shall say that X is Π-elliptically admissible if the following conditions hold: (a) X admits a k-core [in the sense of [Mzk6], Remark 2.1.1] X → C; (b) C is semi-elliptic [cf. §0], hence admits a double covering D → C by a once-punctured elliptic curve D; (c) X admits a ﬁnite étale covering Y → X by a hyperbolic curve Y over a ﬁnite extension k Y of k that arises from a normal open subgroup Π Y ⊆ Π such that the resulting ﬁnite étale covering Y → C factors as the composite of a covering Y → D with the covering D → C and, moreover, is such that, for every set of primes Σ such that some open subgroup of Δ is pro-Σ, it holds that Δ is pro-Σ, and, moreover, the degree of the covering Y → C × k k Y is a product of primes [perhaps with multiplicities] ∈ Σ. When Π = π 1 (X), we shall simply say that X is elliptically admissible. Remark 3.1.1. In the notation of Deﬁnition 3.1, one veriﬁes immediately that D k → C k may be characterized as the unique [up to isomorphism over C k ] ﬁnite étale double covering of C k by a hyperbolic curve [i.e., as opposed to an arbitrary hyperbolic orbicurve]. 66 SHINICHI MOCHIZUKI Example 3.2. Scheme-theoretic Elliptic Cuspidalizations. (i) Let N be a positive integer; D a once-punctured elliptic curve over a ﬁnite Galois extension k of k such that all of the N -torsion points of the underlying elliptic curve E of D are deﬁned over k ; D → C a semi-elliptic k -core of D [such that D → C is the double covering appearing in the deﬁnition of “semi-elliptic”]. Then the morphism [N ] E : E → E given by multiplication by N determines a ﬁnite étale covering [N ] D : U → D [of degree N 2 ], together with an open embedding U → D [which we use to identify U with its image in D], i.e., we have a diagram as follows: U → D ⏐ ⏐ [N ] D D Suppose that the Galois group Gal(k/k) is slim. Then, in the language of [Mzk15], §4, this situation may be described as follows [cf. [Mzk15], Deﬁnition 4.2, (i), where we take the extension “1 → Δ → Π → G → 1” to be the extension 1 → π 1 (D × k k) → π 1 (D) → Gal(k/k ) → 1]: The above diagram yields a chain D U (→ D) (U →) U n (U n →) U n−1 . . . def (U 3 →) U 2 (U 2 →) U 1 = D def [where n = N 2 − 1] whose associated type-chain is , •, . . . , • [i.e., a ﬁnite étale covering, followed by n de-cuspidalizations], together with a terminal isomorphism ∼ U 1 → D [which, in our notation, amounts to the identity morphism] from the U 1 at the end of the above chain to the unique D of the trivial chain [of length 0]. In particular: The above chain may thought of as a construction of a “cuspidalization” [i.e., result of passing to an open subscheme by removing various closed points] U → D of D. The remainder of the portion of the theory of the present §3 concerning elliptic cuspidalizations consists, in essence, of the unraveling of various consequences of this “chain-theoretic formulation” of the diagram that appears at the beginning of the present item (i). (ii) A variant of the discussion of (i) may be obtained as follows. In the notation of (i), suppose further that X is an elliptically admissible hyperbolic orbicurve over k, and that we have been given ﬁnite étale coverings V → X, V → D, where V is a hyperbolic curve over k . Also, [for simplicity] we suppose that V → X is a Galois def covering such that Gal(V /X) preserves the open subscheme U V = V × D U ⊆ V [i.e., the inverse image of U ⊆ D via V → D]. Thus, U V ⊆ V descends to an open subscheme U X ⊆ X. Then by appending to the chain of (i) the “ﬁnite étale TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 67 covering” V → X, followed by the “ﬁnite étale quotient” V → D on the left, and the “ﬁnite étale covering” V → D, followed by the “ﬁnite étale quotient” V → X on the right, we obtain a chain X V (→ X) (V →) D U (→ D) (U →) U n (U n →) U n−1 . . . def def (U 3 →) U 2 (U 2 →) U 1 = D V (→ D) (V →) X ∗ = X whose associated type-chain is , , , •, . . . , •, , ∼ [where the “. . . ” are all “•’s”], together with a terminal isomorphism X ∗ → X [i.e., the identity morphism]. In particular, the above chain may thought of as a construction of a “cuspidalization” U X → X of X via the construction of a “cuspi- dalization” U V → V of V , equipped with descent data [i.e., a suitable collection of automorphisms] with respect to the ﬁnite étale Galois covering V → X. Now by translating the scheme-theoretic discussion of Example 3.2 into the language of proﬁnite groups via the theory of [Mzk15], §4, we obtain the following result. Corollary 3.3. (Pro-Σ Elliptic Cuspidalization I: Algorithms) Let D be a chain-full set of collections of partial construction data [cf. [Mzk15], Deﬁnition 4.6, (i)] such that the rel-isom-DGC holds [i.e., the “relative isomorphism version of the Grothendieck Conjecture for D holds” — cf. [Mzk15], Deﬁnition 4.6, (ii)]; G a slim proﬁnite group; 1 → Δ → Π → G → 1 an extension of GSAFG-type that admits partial construction data (k, X, Σ), where k is of characteristic zero, and X is a Π-elliptically admissible [cf. Def- inition 3.1] hyperbolic orbicurve, such that ([X], [k], Σ) ∈ D; α : π 1 (X) Π the corresponding scheme-theoretic envelope [cf. [Mzk15], Deﬁnition 2.1, (iii)]; ∼ → X the pro-ﬁnite étale covering of X determined by α [so Π → Gal( X/X)]; X ∼ k the resulting ﬁeld extension of k [so G → Gal( k/k)]. Suppose further that, for × some l ∈ Σ, the cyclotomic character G → Z l has open image. Thus, by the theory of [Mzk15], §4, we have associated categories Chain(Π); Chain iso-trm (Π); ÉtLoc(Π) which may be constructed via purely “group-theoretic” operations from the extension of proﬁnite groups 1 → Δ → Π → G → 1 [cf. [Mzk15], Deﬁnition 4.2, (iii), (iv), (v); [Mzk15], Lemma 4.5, (v); the proof of [Mzk15], Theorem 4.7, (ii)]. Then: (i) Let G ⊆ G be a normal open subgroup, corresponding to some ﬁnite ex- def def k of k; Π = Π × G G ; C a k -core of X k = X × k k . Then tension k ⊆ the ﬁnite étale covering X k → C determines a chain X k C of the category k ) [cf. [Mzk15], Deﬁnition 4.2, (i), (ii)] whose image Π Π C in Chain( X/X Chain(Π ) [via the natural functor of [Mzk15], Remark 4.2.1] may be characterized 68 SHINICHI MOCHIZUKI “group-theoretically”, up to isomorphism in Chain(Π ), as the unique chain of length 1 in Chain(Π ), with associated type-chain , such that the resulting object of ÉtLoc(Π ) forms a terminal object of ÉtLoc(Π ). (ii) The collection of open subgroups Π D ⊆ Π C that arise from ﬁnite étale double coverings D → C that exhibit C as semi-elliptic [cf. Remark 3.1.1] may be characterized “group-theoretically” as the collection of open subgroups def J ⊆ Π C of index 2 such that J Δ C [where Δ C = Ker(Π C G )] is torsion-free [i.e., the covering determined by J is a scheme — cf. [Mzk15], Lemma 4.1, (iv)]. def (iii) Write Δ D = Ker(Π D G ). Let N be a positive integer which is a product of primes [perhaps with multipliticites] of Σ; U ⊆ D the open subscheme obtained by removing the N -torsion points of the elliptic curve underlying D; V → X, V → D ﬁnite étale coverings, where V is a hyperbolic curve over k . Suppose further that V → X arises from a normal open subgroup Π V ⊆ Π such that def Gal(V /X) ∼ = Π/Π V preserves the open subscheme U V = V × D U ⊆ V [i.e., the inverse image of U ⊆ D via V → D], while V → D arises from an open immersion Π V → Π D . Thus, U V ⊆ V descends to an open subscheme U X ⊆ X, and U ⊆ D, U V ⊆ V , U X ⊆ X determine extensions of GSAFG-type 1 → Δ U → Π U → G → 1; 1 → Δ U V → Π U V → G → 1 1 → Δ U X → Π U X → G → 1 [i.e., by considering the ﬁnite étale Galois coverings of degree a product of primes [perhaps with multipliticites] ∈ Σ over coverings of U , U V , U X arising from Π], together with natural surjections Π U Π D , Π U V Π V , Π U X Π and open im- mersions Π U V → Π U , Π U V → Π U X . [In particular, Δ U , Δ U V , and Δ U X are pro-Σ groups.] Then, for any G ⊆ G that is suﬃciently small, where “suﬃciently” depends only on N , the natural surjection Π U X Π — i.e., “cuspidalization” of Π — may be constructed via “group-theoretic” operations as follows: (a) There exists a [not necessarily unique] Π-chain, which admits an en- tirely “group-theoretic” description, with associated type-chain , , , •, . . . , •, , — cf. Example 3.2, (ii) — that admits a terminal isomorphism with the trivial Π-chain [of length 0], and whose ﬁnal three groups consist of Π D Π V (→ Π D ) (Π V →) Π such that the natural surjection Π U Π D may be recovered from the chain of “•’s” terminating at the third to last group of the above-mentioned Π-chain; the natural surjection Π U V Π V may then be recovered from Π U Π D by forming the ﬁber product with the inclusion Π V → Π D . (b) The natural surjection Π U X Π may be recovered from Π U V Π V [where we note that Π U V Π V may be identiﬁed with the ﬁber product of TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 69 out Π U X Π with the inclusion Π V → Π] by forming the “ ” [cf. §0] with respect to the unique lifting [relative to Π U V Π V ] of the outer action of the ﬁnite group Π/Π V on Π V to a group of outer automorphisms of Π U V . (c) The decomposition groups of the closed points of X lying in the complement of U X may be obtained as the images via Π U X Π of the cuspidal decomposition groups of Π U X [cf. [Mzk15], Lemma 4.5, (v)]. Proof. The assertions of Corollary 3.3 follow immediately from the deﬁnitions, together with the various references quoted in the course of the “group-theoretic” reconstruction algorithm described in the statement of Corollary 3.3, and the equiv- alences of [Mzk15], Theorem 4.7, (i). Remark 3.3.1. Let p be a prime number. Then if one takes F to be set of iso- morphism classes of generalized sub-p-adic ﬁelds, S the set of sets of prime numbers containing p, and V to be the set of isomorphism classes of hyperbolic orbicurves def over ﬁelds whose isomorphism class ∈ F, then D = V×F×S satisﬁes the hypothesis of Corollary 3.3 concerning “D” [cf. [Mzk15], Example 4.8, (i)]. Remark 3.3.2. Recall that when k is an MLF or an NF, the subgroup Δ ⊆ Π admits a purely “group-theoretic” characterization [cf. [Mzk15], Theorem 2.6, (v), (vi)]. Thus, when k is an MLF or an NF, the various “group-theoretic” reconstruc- tion algorithms described in the statement of Corollary 3.3 may be thought of as being applied not to the extension 1 → Δ → Π → G → 1, but rather to the single proﬁnite group Π. Remark 3.3.3. One veriﬁes immediately that Corollary 3.3 admits a “tempered version”, when the base ﬁeld k is an MLF [cf. [Mzk15], Theorem 4.12, (i)]. We leave the routine details to the reader. Remark 3.3.4. By applying the tempered version of Corollary 3.3 discussed in Remark 3.3.3, one may obtain “explicit reconstruction algorithm versions” of certain results of [Mzk12] [cf. [Mzk12], Theorem 1.6; [Mzk12], Remark 1.6.1] concerning the étale theta function. We leave the routine details to the reader. The “group-theoretic” algorithm of Corollary 3.3 has the following immediate “Grothendieck Conjecture-style” consequence. Corollary 3.4. (Pro-Σ Elliptic Cuspidalization II: Comparison) Let D be a chain-full set of collections of partial construction data [cf. [Mzk15], Deﬁnition 4.6, (i)] such that the rel-isom-DGC holds [cf. [Mzk15], Deﬁnition 4.6, (ii)]. For i = 1, 2, let G i be a slim proﬁnite group; 1 → Δ i → Π i → G i → 1 70 SHINICHI MOCHIZUKI an extension of GSAFG-type that admits partial construction data (k i , X i , Σ i ), where k i is of characteristic zero, and X i is a Π i -elliptically admissible [cf. Deﬁnition 3.1] hyperbolic orbicurve, such that ([X i ], [k i ], Σ i ) ∈ D; α i : π 1 (X i ) Π i the corresponding scheme-theoretic envelope [cf. [Mzk15], Deﬁ- i → X i the pro-ﬁnite étale covering of X determined by α i nition 2.1, (iii)]; X ∼ ∼ i /X i )]; [so Π i → Gal( X k i the resulting ﬁeld extension of k i [so G i → Gal( k i /k i )]; C i a k i -core of X i ; D i → C i a ﬁnite étale double covering that exhibits C i as semi-elliptic [cf. Remark 3.1.1]; Π i ⊆ Π C i , Π D i ⊆ Π C i the open subgroups deter- integer which is a product of primes mined by X i → C i , D i → C i ; N a positive [perhaps with multipliticites] ∈ Σ 1 Σ 2 ; U i ⊆ D i the open subscheme obtained by removing the N -torsion points of the elliptic curve underlying D i ; V i → X i , V i → D i ﬁnite étale coverings that arise from a normal open subgroup Π V i ⊆ Π i and an open immersion Π V i → Π D i such that Gal(V i /X i ) ∼ = Π i /Π V i preserves def the open subscheme U V i = V i × D i U i ⊆ V i [i.e., the inverse image of U i ⊆ D i via V i → D i ]; U X i ⊆ X i the resulting open subscheme [obtained by descending U V i ⊆ V i ]; 1 → Δ U Xi → Π U Xi → G i → 1 the extension of GSAFG-type obtained [via α i ] by considering the ﬁnite étale Galois coverings of degree a product of primes [perhaps with multipliticites] ∈ Σ i over coverings of U X i arising from Π i ; Π U Xi Π i the natural surjection [relative to α i ]. Suppose further that, for some l ∈ Σ 1 Σ 2 , the cyclotomic characters G i → Z × l have open image for i = 1, 2. Let ∼ φ : Π 1 → Π 2 be an isomorphism of proﬁnite groups such that φ(Δ 1 ) = Δ 2 . Then there exists an isomorphism of proﬁnite groups ∼ φ U : Π U X 1 → Π U X 2 that is compatible with φ, relative to the natural surjections Π U Xi Π i . More- over, such an isomorphism is unique up to composition with an inner automor- phism arising from an element of the kernel of Π U Xi Π i . Proof. The construction of φ U follows immediately from Corollary 3.3; the asserted uniqueness then follows immediately from our assumption that the rel-isom-DGC holds. Remark 3.4.1. Just as in the case of Corollary 3.3 [cf. Remark 3.3.3], Corollary 3.4 admits a “tempered version”, when the base ﬁelds k i invoved are MLF’s. We leave the routine details to the reader. Remark 3.4.2. By applying Corollary 3.4 [cf. also Remarks 3.3.4, 3.4.1], one may obtain “pro-Σ tempered” versions of certain results of [Mzk12] [cf. [Mzk12], Theorem 1.6; [Mzk12], Remark 1.6.1] concerning the étale theta function. We leave the routine details to the reader. TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 71 Now we return to the notation introduced at the beginning of the present §3: Let X be a hyperbolic orbicurve over a ﬁeld k of characteristic zero; k an algebraic closure of k. Thus, we have an exact sequence of fundamental groups 1 → π 1 (X × k k) → π 1 (X) → Gal(k/k) → 1. Deﬁnition 3.5. We shall say that X is of strictly Belyi type if it is deﬁned over a number ﬁeld and isogenous [cf. §0] to a hyperbolic curve of genus zero. [Thus, this deﬁnition generalizes the deﬁnition of [Mzk14], Deﬁnition 2.3, (i).] Example 3.6. Scheme-theoretic Belyi Cuspidalizations. (i) Let P be a copy of the projective line minus three points over a ﬁnite Galois extension k of k; V an arbitrary hyperbolic curve over k ; U ⊆ V a nonempty open subscheme [hence, in particular, a hyperbolic curve over k ]. Suppose that U [hence also V ] is deﬁned over a number ﬁeld. Then it follows from the existence of Belyi maps [cf. [Belyi]; [Mzk7]] that, for some nonempty open subscheme W ⊆ U , there exists a diagram as follows: W → ⏐ ⏐ β U → V P [where the “→’s” are the natural open immersions; the “Belyi map” β is ﬁnite étale]. By replacing k by some ﬁnite extension of k , let us suppose further [for simplicity] that the cusps of W are all deﬁned over k . Also, let us suppose that the Galois group Gal(k/k) is slim. Then, in the language of [Mzk15], §4, this situation may be described as follows [cf. [Mzk15], Deﬁnition 4.2, (i), where we take the extension “1 → Δ → Π → G → 1” to be the extension 1 → π 1 (P × k k) → π 1 (P ) → Gal(k/k ) → 1]: For some nonnegative integers n, m, the above diagram yields a chain P W (→ P ) (W →) W n (W n →) W n−1 . . . def (W 2 →) W 1 = U (U →) U m (U m →) U m−1 . . . def (U 2 →) U 1 = V whose associated type-chain is , •, . . . , • [i.e., a ﬁnite étale covering, followed by n + m de-cuspidalizations]. In particular: The above chain may thought of as a construction of a “cuspidalization” [i.e., result of passing to an open subscheme by removing various closed points] U → V of V . The remainder of the portion of the theory of the present §3 concerning Belyi cuspidalizations consists, in essence, of the unraveling of various consequences of 72 SHINICHI MOCHIZUKI this “chain-theoretic formulation” of the diagram that appears at the beginning of the present item (i). (ii) A variant of the discussion of (i) may be obtained as follows. In the notation of (i), suppose further that X is a hyperbolic orbicurve of strictly Belyi type over k, and that we have been given ﬁnite étale coverings V → X, V → Q, together with an open immersion Q → P [so Q is a hyperbolic curve of genus zero over k ]. Also, [for simplicity] we suppose that V → X is Galois, that U ⊆ V descends to an open subscheme U X ⊆ X, and [by possibly replacing k by a ﬁnite extension of k ] that the cusps of Q are deﬁned over k . Then by appending to the chain of (i) the “ﬁnite étale covering” V → X, followed by the “ﬁnite étale quotient” V → Q, followed by def the de-cuspidalizations Q → Q l → . . . → Q 1 = P [for some nonnegative integer l], on the left, and the “ﬁnite étale quotient” V → X on the right, we obtain a chain def X V (→ X) (V →) Q (Q →) Q l . . . (Q 2 →) Q 1 = P def W (→ P ) (W →) W n . . . (W 2 →) W 1 = U (U →) U m . . . def def (U 2 →) U 1 = V (V →) X ∗ = X whose associated type-chain is , , •, . . . , •, , •, . . . , •, ∼ [where the “. . . ” are all “•’s”], together with a terminal isomorphism X ∗ → X [i.e., the identity morphism]. In particular, the above chain may thought of as a construction of a “cuspidalization” U X → X of X via the construction of a “cuspidalization” U → V of V , equipped with descent data [i.e., a suitable collection of automorphisms] with respect to the ﬁnite étale Galois covering V → X. Now by translating the scheme-theoretic discussion of Example 3.6 into the language of proﬁnite groups via the theory of [Mzk15], §4, we obtain the following result. Corollary 3.7. (Proﬁnite Belyi Cuspidalization I: Algorithms) Let D be a chain-full set of collections of partial construction data [cf. [Mzk15], Deﬁnition 4.6, (i)] such that the rel-isom-DGC holds [i.e., the “relative isomorphism version of the Grothendieck Conjecture for D holds” — cf. [Mzk15], Deﬁnition 4.6, (ii)]; G a slim proﬁnite group; 1 → Δ → Π → G → 1 an extension of GSAFG-type that admits partial construction data (k, X, Σ), where k is of characteristic zero, X is a hyperbolic orbicurve of strictly Belyi type [cf. Deﬁnition 3.5], and Σ is the set of all primes, such that ∼ ([X], [k], Σ) ∈ D; α : π 1 (X) → Π the corresponding scheme-theoretic envelope [cf. [Mzk15], Deﬁnition 2.1, (iii)], which is an isomorphism of proﬁnite groups; ∼ → X the pro-ﬁnite étale covering of X determined by α [so Π → Gal( X/X)]; X ∼ k the resulting algebraic closure of k [so G → Gal( k/k)]. Suppose further that, for × some l ∈ Σ, the cyclotomic character G → Z l has open image. Thus, by the theory of [Mzk15], §4, we have associated categories Chain(Π); Chain iso-trm (Π); ÉtLoc(Π) TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 73 which may be constructed via purely “group-theoretic” operations from the extension of proﬁnite groups 1 → Δ → Π → G → 1 [cf. [Mzk15], Deﬁnition 4.2, (iii), (iv), (v); [Mzk15], Lemma 4.5, (v); the proof of [Mzk15], Theorem 4.7, (ii)]. Then for every nonempty open subscheme U X ⊆ X deﬁned over a number ﬁeld, the natural surjection def ∼ Π U X = π 1 (U X ) π 1 (X) → Π ∼ [where the ﬁnal “ → ” is given by the inverse of α] — i.e., “cuspidalization” of Π — may be constructed via “group-theoretic” operations as follows: (a) For some normal open subgroup Π V ⊆ Π, which corresponds to a ﬁ- nite covering V → X of hyperbolic orbicurves, there exists a [not neces- sarily unique] Π-chain, which admits an entirely “group-theoretic” description, with associated type-chain , , •, . . . , •, , •, . . . , •, — cf. Example 3.6, (ii) — that admits a terminal isomorphism with def the trivial Π-chain [of length 0] such that if we write U = V × X U X , def Π U = Π V × Π Π U X , then the natural surjection Π U Π V may be recovered from the chain of “•’s” terminating at the second to last group of the above-mentioned Π-chain. (b) The natural surjection Π U X Π may be recovered from Π U Π V out by forming the “ ” [cf. §0] with respect to the unique lifting [relative to Π U Π V ] of the outer action of the ﬁnite group Π/Π V on Π V to a group of outer automorphisms of Π U . (c) The decomposition groups of the closed points of X lying in the complement of U X may be obtained as the images via Π U X Π of the cuspidal decomposition groups of Π U X [cf. [Mzk15], Lemma 4.5, (v)]. Proof. The assertions of Corollary 3.7 follow immediately from the deﬁnitions, together with the various references quoted in the course of the “group-theoretic” reconstruction algorithm described in the statement of Corollary 3.7, and the equiv- alences of [Mzk15], Theorem 4.7, (i). Remark 3.7.1. Similar remarks to Remarks 3.3.1, 3.3.2, 3.3.3 may be made for Corollary 3.7. Remark 3.7.2. In the situation of Corollary 3.7, when the ﬁeld k is an MLF, one then obtains an algorithm for constructing the decomposition groups of arbitrary closed points of X, by combining the algorithms of Corollary 3.7 — cf., especially, 74 SHINICHI MOCHIZUKI Corollary 3.7, (c), which allows one to construct the decomposition groups of those closed points of X which [like U X !] are deﬁned over a number ﬁeld — with the “p- adic approximation lemma” of [Mzk8] [i.e., [Mzk8], Lemma 3.1]. A “Grothendieck Conjecture-style” version of this sort of reconstruction of decomposition groups of arbitrary closed points of X may be found in [Mzk8], Corollary 3.2. The “group-theoretic” algorithm of Corollary 3.7 has the following immediate “Grothendieck Conjecture-style” consequence. Corollary 3.8. (Proﬁnite Belyi Cuspidalization II: Comparison) Let D be a chain-full set of collections of partial construction data [cf. [Mzk15], Deﬁnition 4.6, (i)] such that the rel-isom-DGC holds [cf. [Mzk15], Deﬁnition 4.6, (ii)]. For i = 1, 2, let G i be a slim proﬁnite group; 1 → Δ i → Π i → G i → 1 an extension of GSAFG-type that admits partial construction data (k i , X i , Σ i ), where k i is of characteristic zero, X i is a hyperbolic orbicurve of strictly Belyi type [cf. Deﬁnition 3.5], and Σ i is the set of all primes, such that ∼ ([X i ], [k i ], Σ i ) ∈ D; α i : π 1 (X i ) → Π i the corresponding scheme-theoretic en- velope [cf. [Mzk15], Deﬁnition 2.1, (iii)], which is an isomorphism of proﬁnite i → X i the pro-ﬁnite étale covering of X i determined by α i [so groups; X ∼ ∼ i /X i )]; Π i → Gal( X k i the resulting algebraic closure of k i [so G i → Gal( k i /k i )]. If, for i = 1, 2, U X i ⊆ X i is a nonempty open subscheme which is deﬁned over a number ﬁeld, then write 1 → Δ U Xi → Π U Xi → G i → 1 for the extension of GSAFG-type determined by [α i and] the natural surjection π 1 (U X i ) Gal( k i /k i ) ( ∼ = G i ); Π U Xi Π i for the natural surjection [relative to α i ]. Suppose further that, for some l ∈ Σ 1 Σ 2 , the cyclotomic characters G i → Z × l have open image for i = 1, 2. Let ∼ φ : Π 1 → Π 2 be an isomorphism of proﬁnite groups such that φ(Δ 1 ) = Δ 2 . Then, for each nonempty open subscheme U X 1 ⊆ X 1 deﬁned over a number ﬁeld, there exist a nonempty open subscheme U X 2 ⊆ X 2 deﬁned over a number ﬁeld and an isomor- phism of proﬁnite groups ∼ φ U : Π U X 1 → Π U X 2 that is compatible with φ, relative to the natural surjections Π U Xi Π i . More- over, such an isomorphism φ U is unique up to composition with an inner auto- morphism arising from an element of the kernel of Π U Xi Π i . Proof. The construction of φ U for a suitable nonempty open subscheme U X 2 follows immediately from Corollary 3.7; the asserted uniqueness then follows im- mediately from our assumption that the rel-isom-DGC holds. Remark 3.8.1. 3.8. A similar remark to Remark 3.4.1 may be made for Corollary TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 75 Bibliography [Belyi] G. V. Belyi, On Galois extensions of a maximal cyclotomic ﬁeld, Izv. Akad. Nauk SSSR Ser. Mat. 43:2 (1979), pp. 269-276; English transl. in Math. USSR- Izv. 14 (1980), pp. 247-256. [BLR] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Ergebnisse der Math- ematik und ihrer Grenzgebiete 21, Springer-Verlag (1990). [Ill] L. Illusie, An Overview of the Work of K. Fujiwara, K. Kato and C. Nakayama on Logarithmic Etale Cohomology, in Cohomologies p-adiques et applications arithmétiques (II), P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport, eds., Asterisque 279 (2002), pp. 271-322. [Kato] K. Kato, Toric Singularities, Amer. J. Math. 116 (1994), pp. 1073-1099. [Knud] F. F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II, Math. Scand. 52 (1983), pp. 161-199. [Kobl] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions, Graduate Texts in Mathematics 58, Springer-Verlag (1977). [Mzk1] S. Mochizuki, The Geometry of the Compactiﬁcation of the Hurwitz Scheme, Publ. Res. Inst. Math. Sci. 31 (1995), pp. 355-441. [Mzk2] S. Mochizuki, Extending Families of Curves over Log Regular Schemes, J. reine angew. Math. 511 (1999), pp. 43-71. [Mzk3] S. Mochizuki, The Local Pro-p Anabelian Geometry of Curves, Invent. Math. 138 (1999), pp. 319-423. [Mzk4] S. Mochizuki, Topics Surrounding the Anabelian Geometry of Hyperbolic Curves, Galois Groups and Fundamental Groups, Mathematical Sciences Research In- stitute Publications 41, Cambridge University Press (2003), pp. 119-165. [Mzk5] S. Mochizuki, The Absolute Anabelian Geometry of Hyperbolic Curves, Galois Theory and Modular Forms, Kluwer Academic Publishers (2004), pp. 77-122. [Mzk6] S. Mochizuki, The Absolute Anabelian Geometry of Canonical Curves, Kazuya Kato’s ﬁftieth birthday, Doc. Math. 2003, Extra Vol., pp. 609-640. [Mzk7] S. Mochizuki, Noncritical Belyi Maps, Math. J. Okayama Univ. 46 (2004), pp. 105-113. [Mzk8] S. Mochizuki, Galois Sections in Absolute Anabelian Geometry, Nagoya Math. J. 179 (2005), pp. 17-45. [Mzk9] S. Mochizuki, Semi-graphs of Anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), pp. 221-322. [Mzk10] S. Mochizuki, Global Solvably Closed Anabelian Geometry, Math. J. Okayama Univ. 48 (2006), pp. 57-71. [Mzk11] S. Mochizuki, Conformal and quasiconformal categorical representation of hy- perbolic Riemann surfaces, Hiroshima Math. J. 36 (2006), pp. 405-441. 76 SHINICHI MOCHIZUKI [Mzk12] S. Mochizuki, The Étale Theta Function and its Frobenioid-theoretic Manifes- tations, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 227-349. [Mzk13] S. Mochizuki, A combinatorial version of the Grothendieck conjecture, Tohoku Math J. 59 (2007), pp. 455-479. [Mzk14] S. Mochizuki, Absolute anabelian cuspidalizations of proper hyperbolic curves, J. Math. Kyoto Univ. 47 (2007), pp. 451-539. [Mzk15] S. Mochizuki, Topics in Absolute Anabelian Geometry I: Generalities, J. Math. Sci. Univ. Tokyo 19 (2012), pp. 139-242. [Mzk16] S. Mochizuki, Topics in Absolute Anabelian Geometry III: Global Reconstruc- tion Algorithms, RIMS Preprint 1626 (March 2008); updated version available at the following web site: http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html [Mzk17] Y. Hoshi and S. Mochizuki, Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and proﬁnite Dehn twists, Galois-Teichmüller Theory and Arithmetic Geometry, Adv. Stud. Pure Math. 63, Math. Soc. Japan (2012), pp. 659-811. [MT] S. Mochizuki, A. Tamagawa, The algebraic and anabelian geometry of conﬁg- uration spaces, Hokkaido Math. J. 37 (2008), pp. 75-131. [SdTm] M. Saidi and A. Tamagawa, A prime-to-p version of Grothendieck’s anabelian conjecture for hyperbolic curves over ﬁnite ﬁelds of characteristic p > 0, Publ. Res. Inst. Math. Sci. 45 (2009), pp. 135-186. [Serre] J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer- Verlag (1973). [Tama1] A. Tamagawa, The Grothendieck Conjecture for Aﬃne Curves, Compositio Math. 109 (1997), pp. 135-194. [Tama2] A. Tamagawa, Resolution of nonsingularities of families of curves, Publ. Res. Inst. Math. Sci. 40 (2004), pp. 1291-1336. [Tate] J. Tate, p-Divisible Groups, Proceedings of a Conference on Local Fields, Driebergen, Springer-Verlag (1967), pp. 158-183.